Combine sun-dried tomatoes, 2 tablespoons olive oil, balsamic vinegar, tomato oil, garlic, and sugar in a small bowl. Toss to coat and set aside.

Use a sharp knife to cut the purple translucent membrane away from each fillet. Cut each fillet crosswise into 3 to 4 pieces. Season with salt and pepper.

Heat 1 tablespoon olive oil and butter in a large skillet over medium-high heat. Sear monkfish until golden brown, about 3 minutes. Flip and continue cooking until opaque throughout, 2 to 3 minutes more. Reduce heat to low, remove fish to a plate, and cover with aluminum foil to keep warm.

Pour sun-dried tomato mixture into the skillet and quickly stir around the pan until just warmed, about 20 seconds. Spoon topping over fish and garnish with parsley.