broil (v.1)
"to cook," late 14c. (earlier "to burn," mid-14c.), from Old French bruller "to broil, roast" (Modern French brûler), earlier brusler "to burn" (11c.), which, with Italian bruciare, is of uncertain and much-disputed origin.

Perhaps from Vulgar Latin *brodum "broth," borrowed from Germanic and ultimately related to brew (v.). Gamillscheg proposes it to be from Latin ustulare "to scorch, singe" (from ustus, past participle of urere "to burn") and altered by influence of Germanic "burn" words beginning in br-. Related: Broiled; broiling.
broil (v.2)
early 15c., "to quarrel, brawl," also "mix up, present in disorder," from Anglo-French broiller "mix up, confuse," Old French brooillier "to mix, mingle," figuratively "to have sexual intercourse" (13c., Modern French brouiller), perhaps from breu, bro "stock, broth, brew," from Frankish or another Germanic source (compare Old High German brod "broth") akin to broth (see brew (v.)); also compare imbroglio.