1. Combine first 3 ingredients and 1/2 teaspoon kosher salt in food processor; pulse until parsley is finely chopped. With processor running, gradually add oil through food chute in a slow, steady stream. Pour herb oil through fine-mesh strainer, discarding solids.
2. Heat 3 tablespoons herb oil and fennel seed in a large, deep skillet or Dutch oven over medium heat. Add bell pepper, onion, 3/4 teaspoon kosher salt, and 1/2 teaspoon black pepper. Cook, stirring often, 25 to 30 minutes (reducing heat if needed) or until vegetables are very tender.
3. Bring balsamic vinegar and sugar to a boil in a small saucepan. Boil gently 2 to 3 minutes or until syrupy and reduced to 3 tablespoons. (Watch carefully during last minute as syrup forms quickly.)
4. Place fillets on an oiled baking sheet. Sprinkle with remaining 1/4 teaspoon kosher salt and 1/2 teaspoon black pepper. Drizzle 2 tablespoons herb oil evenly on fillets. Roast in center of oven 8 to 10 minutes or until fish flakes with a fork. Spoon peperonata (vegetables) onto 4 plates. Top with fillets. Drizzle remaining herb oil and 1/2 to 1 teaspoon balsamic syrup over vegetables and fish on each plate.