Fettunta is a contraction of the Italian words fetta and unta, often (awkwardly) translated as ‘oily slice’ in English. Una fetta is indeed a slice of something—cake, bread, cheese, meat—and unta does in fact mean ‘oily.’ But I think we can do better.
Fetta is related to the Italian verb affettare: to cut or to slice (a deli-style meat slicer is called un’affettatrice; to cut oneself is affettarsi). Then, unto/a is the past participle of the verb ungere, which means to oil, grease, or anoint. The English words unguent and unction have the same Latin root (yet are rarely used in food-related contexts, one hopes).
Back to fettunta. Slice a thick fetta of Tuscan bread and grill it on both sides. Rub down both sides with a peeled clove of garlic. Drizzle both sides with a really good new olive oil—this is the unta part—and dust with salt. Use your fingers to rub the oil and salt all over the bread. Put lots of napkins out. Buona fettunta!