This kind of word-boundary reanalysis is very common across languages. In English, it has happened many, many times when words that used to be vowel-initial acquired an initial /n/, or lost it, because of the preceding indefinite article:

However in this case, since the following word started with /br/, the effect of M-mobile was not to add an /m-/ to the following word, but to reduce the cluster between the nasal & trilled cons. Across lgs, this precise context is acoustically very unstable:

Other words in the history of Georgian have undergone this same reanalysis; a preceding demonstrative like იმ im 'that' or ამ am 'this' was reinterpreted by speakers as part of the following noun: Old Georgian იმ ტრედისა im ṭredisa > Modern Georgian იმ მტრედისა im mṭredisa

So a semantic shift from a word used to describe roasting grains is not so surprising. The precise source of this loan word is less than 100% clear, but it probably came from one of a number of northwestern Iranian languages with something like the form /brej-/:

Weekly Georgian Etymology: გარუჯული garujuli 'sun-tanned', from Middle Georgian რუჯვილი rujvili 'roasted', reanalysis of *brujvili, loan of a northwest Iranian root brej- roast, from Indo-European *bʰreǵ- roast. Cognate with Greek φρύγω roast, Latin frīgō roast, and English fry.

Weekly Georgian Etymology: სტუმარი sṭumari 'guest', from Old Georgian ႱႲႳႫႠႰႨ sṭumari, possibly an ancient compound-loan of Nakh *psṭu wife and *mar husband, itself a loan from Kartvelian *kmarʲ- husband. It reflects a semantic shift from a kinship term to one of social bonds.

This Hurrian root is in fact one of the earliest attested roots in that language -- it is found on the very earliest Hurrian text dedicated to a king named Tish-atal (or Tish-adal) who lived around 2000 BC in Urkesh and whose dedicated statue is now found in the Louvre:

One possible loan source is Hurrian 𒀀𒁕𒀠 adal- strong. Hurrian was spoken in eastern Anatolia and northern Syria in the Bronze Age, and along with Assyrian, its speakers would have been in contact with the earliest Kartvelians through trading networks called karū (emporia):

Weekly Georgian Etymology: ძლევა dzleva 'victory; overcoming', from Old Georgian ႻႪႤႥႠ dzleva, from Georgian-Zan *dzʲal- strong, perhaps an ancient loan from Hurrian 𒀀𒁕𒀠 adal- strong. Also found in the Georgian motto ძალა ერთობაშია Dzala ertobašia 'Strength is in unity'.

I think the Handbook of Languages of the Caucasus says it best: the most realistic analysis for Kabardian, Ubykh and related languages is that there are at least two phonemic vowels.

It's possible this root is a borrowing from the older Nakh-Daghestanian root *jĕrƛ̣w- belt: Chechen döχka Chadakolob ráƛ̣ič Akhvakh raƛ̣:iča Andi roƛ̣:očul Hunzib ɔƛe If so, Kartvelian borrowed the extremely rare Daghestanian sound *ƛ̣ as /ṭq̇/ - very few lgs have this sound.

Which translates:

In its current meaning of 'striking', the word is first attested in a late 16th century Georgian poetic translation of the great Persian epic, The Shahnameh, in which Jimshed theatrically rejects an envoys demands by striking his hat against the ground:

Weekly Georgian Etymology: დარტყმა darṭq̇ma 'kick, strike, hit', derived from Old Georgian ႠႰႲႷႫႠ arṭq̇ma, gird, swipe from Kartvelian *raṭq̇-/*rṭq̇- gird, buckle, loan from or into Nakh-Daghestanian *jĕrƛ̣w- belt, girdle. Semantically shifted from a tool to a verb of motion.

Are the Kartvelian and Nakh-Daghestanian words related? It's hard to say. ND is generally believed to be a somewhat older language family, generally reconstructed to around the middle of the 4th millennium BC, around 1000 to 2000 years before Proto-Kartvelian.

In modern Georgian this word is most often used to refer to a goalkeeper in various forms of sports. In association football (ფეხბურთი pexburti) it first found this meaning in 1911 when that sport was gaining popularity around Georgia.

Weekly Georgian Etymology: მეკარე meḳare 'goalkeeper', from კარი ḳari gate, door, from Old Georgian ႩႠႰႨ ḳari, from Kartvelian *ḳar- open, ancient loan out of / into Nakh-Daghestanian *kɨrṭV- open: Avar karáṭ, Tindi karṭi, Karata karṭa, all 'hole'. Common semantic shift.

So the modern Georgian word for sunflower ultimately goes back to the Kartvelian root for the sun and an ancient Akkadian or other Semitic loanword referring to someone who oversees or watches and protects.

Its late attestation in Georgian has a simple explanation: common sunflowers are native to North America, and did not exist in Georgia before the Columbian Exchange in the 17th century. Around that time, dozens of nearby languages calqued something like 'sun-watcher' for this plant.

The specific form მზესუმზირა mzesumzira lit. 'sun-gazer' is a relatively recent compound form attested no earlier than the 19th century. It exists alongside other similar compounds, e.g. მზეჭვრიტა mze-č̣vriṭa sun-peeper and მზემზირა mze-mzira sun-watcher.

Weekly Georgian Etymology: მზესუმზირა mzesumzira 'sunflower', lit. sun-watcher, compound from ႫႦႨႰႠ mzira gaze, from Georgian-Zan *medzʲir- watch, reanalysis of Akkadian 𒇽𒂗𒉣 maṣṣarum watchman, overseer, agentive form of 𒈾𒍝𒊒 naṣāru guard, from Semitic *NṢR protect.

So the story in short: a Megrelian dish გებჟალია gebžalia referring to flattened ricotta-like cheese stuffed with curds, yogurt and mint comes from the Megrelian word for milk ბჟა bža, which itself has a clear albeit complicated phonological origin in Kartvelian *ɬdzʲe milk.

Where did this initial /b/ come from? The answer: Megrelian also had a special rule that inserted nasal consonants in accented syllables. When /m/ was followed by certain consonants, an epenthetic /b/ was inserted, and sometimes the /m/ was deleted: *ja > *mja > *mbja > bja

This reflects the fact that although Georgians have long had various recipes for preparing spiced or flavored cheeses - several are found in Barbare Jorjadze's 1874 cookbook - this dish, by this name, was a very localized dish characteristic of the Samegrelo region.

გებჟალია gebžalia is not attested in Georgian-language popular cooking and gastronomic materials until very, very recently. It is in fact hardly attested at all before the 2001 Megrelian-German dictionary by Kadschaia:

Weekly Georgian Etymology: გებჟალია gebžalia, cheese roll filled with minty cheese curd, spicy green ajika or yogurt. From the participle of Megrelian გებჟალაფა gebžalapa to milk out, from Megrelian ბჟა bža milk, from Kartvelian *ɬdzʲe milk. A classic west Georgian dish.

The word in this precise form is first used in early 20th century authors arguing for the liberation of oppressed peoples and social classes, e.g. in a 1906 translation of German Marxist theorist Karl Kautsky:

Weekly Georgian Etymology: თვითგამორკვევა tvitgamorḳveva 'self-determination', modern coinage from გამორკვევა gamorḳveva determine, research from Old Georgian ႢႠႰႩႥႤႥႠ garḳveva investigate, discern, clean, from Kartvelian *rḳw- clean, polish. Long part of political discourse.

So the word had probably been in use at least since the early 19th c. The Azerbaijani dish from which the Georgian dish gets its name is quite different in many respects: it is more of a pan-stewed frittata-like dish. This makes sense of the its name, meaning roughly 'sizzler'.

Another piece of evidence that this word existed some time before is that, elsewhere, Jorjadze also uses the word as a verb ჩაჩიხირთმავება čačixirtmaveba in the preparation of a completely different dish of roast rabbit: