1. Fill roasting pan with water. Add cedar plank; weigh down with can or small pot to keep submerged and soak 1 day.
2. Prepare barbecue (high heat). Brush grill rack with 1 tablespoon oil. Arrange tomatoes, onion, and bell pepper on rimmed baking sheet. Sprinkle with salt and pepper; drizzle with 1 tablespoon oil. Grill vegetables until lightly charred and blistered, turning occasionally with tongs, about 4 minutes for tomatoes and 8 minutes for onion and pepper. Return vegetables to same sheet and cool slightly.
3. Peel tomato halves; peel pepper halves. Chop all vegetables coarsely and place in medium bowl. Add olives, anchovies, garlic, basil, capers, vinegar, and remaining 1 tablespoon oil. Toss to blend. Season relish to taste with salt and pepper. (Can be made 1 hour ahead. Let stand at room temperature.)
4. Sprinkle fillets with salt and pepper. Wrap 2 prosciutto slices around each fillet, leaving top and bottom exposed. Secure prosciutto with toothpicks. Press each fillet slightly to flatten top.
5. Drain cedar plank. Place fillets on plank. Spoon 1/4 cup relish atop each fillet and press to adhere. Place plank on grill; cover barbecue. Cook fish until just opaque in center and thermometer inserted into center registers 120°F, checking occasionally and spraying plank with water if beginning to burn, about 18 minutes. Transfer fish to plates. Serve, passing remaining relish.
6. Shopping tip: Use a 1-inch-thick cedar plank for the monkfish. Thinner planks have a tendency to burn and warp on the grill. Look for cedar marked untreated at lumberyards and some home-improvement stores. Or order planks from Satuee Cedar Company ( ; 866-728-8332).