Prepare the vinaigrette. Char the jalapeños directly on a grill, gas stove burner or on a hot griddle pan. Remove jalapeños, place in a small bowl and cover with plastic wrap or a plate. Let steam for about 5minutes. Using a paper towel wipe off charred skin, slice in half, remove seeds and finely dice.
In a food processor add the jalapeños, onion, garlic, mustard, vinegar, salt & pepper, and blend until combined. With the motor running, slowly add the oil and blend until emulsified. Add the basil and blend to combine.
Preheat the grill. Heat grill to medium-high heat (approximately 400°F).
Prepare the fish. Rinse the branzino under cold water and pat dry using paper towels.
Brush both sides of the fish with the oil and season inside and out with the salt and the pepper. Stuff the cavity of each fish with some of the basil and three lemon slices to infuse extra flavor during grilling.
Using butchers’ twine, tie the fish closed around the open cavity.
Grill the branzino. Grill the fish until slightly charred on 1 side, about 5 to 7 minutes. Using a fish or wide spatula, carefully turn the delicate fish over, close the cover and continue grilling until just cooked through, about 6 to 7 minutes more or until the flesh is opaque and easily flakes with a fork. Cooking times may vary depending on the size of the fish, so keep a close eye on them to prevent overcooking
Serve and enjoy. Remove the grilled branzino from the grill and transfer them to a serving platter. Drizzle the jalapeño-basil vinaigrette generously over the grilled fish. Serve immediately with additional lemon wedges and a bowl of the vinaigrette.