How to Make It
Dissolve the yeast in a bowl with the water and sugar. Set aside for 15 minutes.
Stir in the flour and olive oil and knead until the dough is smooth and elastic, about 2 minutes. Cover and let rise in a warm place for 1 hour.
Remove any bones from the anchovy fillets and chop them. Knead the anchovies into the dough. On a floured surface, roll the dough out into a thin sheet about 1/4 inch (6 mm) thick. Use a 2 1/2-inch (6 cm) round cutter to cut into rounds. Re-roll any scraps and cut out more rounds.
Heat the oil in a 12-inch (30 cm) frying pan over medium-high heat until hot. Working in batches, shallow-fry half the dough rounds until golden brown, 1–1 1/2 minutes per side. Repeat with the remaining rounds, adding more oil if necessary. Remove the gonfietti and drain them thoroughly on a plate lined with paper towels. Serve hot.