An omelette is named after a sword blade. Getting from one to the other took a mishearing, a swapped suffix, two letters changing places and an egg.
The starting point is Latin lamella, "a small thin plate," a diminutive of lamina, "plate" or "layer." In Old French this became lemelle, a thin blade. Speakers heard la lemelle, "the blade," as l'alemelle, and the new form alemelle stuck. Linguists call this kind of slip a misdivision: the boundary between article and noun moved.
By the late 1300s, alemelle and its relative alumelle were also being used for a flat dish of beaten eggs, presumably because it was thin and flat like a blade. The cookbook Le Ménagier de Paris, written around 1393, uses these forms. It also calls one such dish an arboulastre, a name Word Histories traces to Latin herba, "herb."
Then the suffix shifted, giving a form alemette, which is not itself recorded. Next two sounds swapped places, a change called metathesis, and alemette became amelette, recorded in 1480. In southern France the first vowel moved from a to o in the mid-1500s. Word Histories suggests this happened under the influence of oeuf, "egg," which would make the egg the last thing to shape the word.
English took the word from French in the early 1600s. Its first known English appearance is in Randle Cotgrave's French-English dictionary of 1611. Before that, Etymonline notes, Middle English had its own term, hanonei, for fried onions mixed with scrambled eggs. Etymonline files the word under the shorter spelling, omelet, and dates it to the 1610s.
The proverb came much later. "You can't make an omelet without breaking eggs" translates a French saying. The French form is recorded by 1721. The earliest English version Word Histories found, from 1796, says "pancakes" instead. Etymonline dates the omelet wording to 1845.
The thin-plate sense survives in the word's relatives. Latin lamina also gave English laminate, first used as a verb in the 1660s.