Preheat the oven to 350°F
Heat a large skillet over high heat with the oil. Add the garlic and cook until golden brown. Add the fennel and a pinch of salt. Cook, stirring, until translucent and fragrant. Add the wine and butter and reduce the heat to a bare simmer. If the liquid is not covering the fennel, add water or vegetable broth to cover. Cover the dish with a lid and simmer until the fennel is cooked through, but has a slight bite, 15-20 minutes.
Place the branzino on a baking sheet, drizzle with olive oil and sprinkle liberally with salt and pepper, including the cavity.
Heat a large ovenproof skillet over high heat. Add the branzino and cook for 3 minutes per side, until the skin is crisp. Transfer to the baking sheet and roast for 5-8 minutes, until cooked through.
In a food processor, blend the roasted red peppers, olive oil, vinegar, salt and pepper until emulsified. Transfer to a bowl and fold in the remaining salsa ingredients.
Divide the braised fennel between plates. Top with branzino and salsa.