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I Fratellini isn't a restaurant, it's a hole-in-the-wall walk-up sandwich and wine bar.
Trattoria ZaZa is near the Mercato Centrale, good food and friendly atmosphere.
Wish I could remember the name (and it's been ~18 years ago) but the first 1-2 blocks across the Ponte Vecchio bridge, away from the "tourist" side and on the "locals", was a fantastic family-style restaurant in the middle of the block, great everything including wine. We were told "everything on that side of the bridge was better"... this place sure was.
We had a great meal at L’Osteria di Giovanni on Via del Moro.