Latin lāmina "thin plate, scale, later, or flake (of metal, etc.)" had the diminutive lāmella, which became Old French lemel(l)e "blade". La lemelle was reanalyzed as l'alemelle, meaning "thin plate, blade of a sword or knife". This was further diminutized to *alemette, which was metathesized to amelette. By this time it meant "omelette", perhaps because of the omelette's thin flat shape. The initial o perhaps came about by association with oeuf "egg".
I learned this from Word Origins and How We Know Them by Anatoly Liberman, a very entertaining and informative read.