Drain baccala and pat dry, checking for bones but leaving intact any skin pieces. Cut baccala into 3-inch by 1-inch pieces and set aside. In a medium mixing bowl, whisk together flour, water, baking powder, 1 teaspoon salt and two tablespoons virgin olive oil until very smooth.
In a 6-inch deep frying pan, heat 1 quart oil to 370 degrees.
Dip baccala into batter and drop into 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 pieces are frying, quarter 3 lemons and remove the seeds. Chop roughly and place in food processor with remaining 1/2 cup of olive oil, teaspoon of salt, sugar and pepper and blend until smooth and thick. Pour lemon pesto into serving bowl. When all baccala is cooked, serve immediately with lemon wedges and lemon pesto dipping sauce.