Drain the baccala and pat dry, checking for bones but leaving intact any pieces of skin. Cut the baccala into 3-inch by 1-inch pieces and set aside. In a medium mixing bowl, whisk together the flour, water, baking powder, 1 teaspoon salt and 2 tablespoons extra-virgin olive oil until very smooth.

In a 6-inch deep frying pan, heat 1-quart oil to 370 degrees F.

Dip the baccala into the batter and drop into the oil, 2 to 3 pieces at a time, and fry until golden brown on all sides, about 6 to 7 minutes. Remove and drain on paper towels. While the pieces are frying, quarter 3 of the lemons and remove the seeds. Chop roughly and place in the food processor with the remaining 1/2-cup if olive oil, teaspoon of salt, sugar and pepper and blend until smooth and thick. Pour the lemon pesto into the serving bowl. When all the baccala is cooked, serve immediately with lemon wedges and lemon pesto dipping sauce.