Translation:Do you want a slice or a bigger piece?
Bucată means both piece and dish though rarely used with the latter sense today from what I've noticed. Both bucată and bucătărie (where dishes are made) come from the same vulgar latin word that meant something like dish.
Silly me, I didn't even notice the similarities between the two words, so thanks for this. Bucată doesn't even need to be food-related, as in the sign "Atenție! Cad bucăți de cărămidă!" that I see here in Timișoara.
So, nothing to do with the latin Buccae for cheek that i know from medicine. i've been using this knowledge as a peg for bucatarie and then piece thinking the latter had to be a mouthful as it was bigger than a slice!
Welp, we have the term bucălat which means having thick/fat cheeks/face or something like that.
Also, biologically, mouth (gură) is cavitate bucală similar to buccal cavity in English.
How do you know that the "mai mare" applies only to the piece and not the slice? How would you say "Do you want a bigger slice or piece" where the word 'bigger' modifies both the slice and the piece?