Bring a large stockpot of water to boil and add 2 tablespoons of salt. Set up an ice bath. Cook the broccoli rabe in boiling water until it is bright green and tender, about 4 minutes. Drain, cool in the ice bath, and drain again. Set aside.
Preheat oven to 375 degrees F.
Lay out 4 (12-inch) pieces of foil out on a counter. Put the fennel, onion, tomatoes, and basil in a medium bowl and season with salt and pepper, to taste. Drizzle with 1 tablespoon of oil and a splash of vinegar. Divide the vegetables between each sheet of foil, making a small mound.
Season the fish on both sides with salt and pepper, to taste, and place 1 on top of each pile of vegetables. Drizzle each pile with a tablespoon of oil and 1 teaspoon of vinegar and use a second piece of foil on each, and roll up the sides, to seal the pouch completely.
Put the pouches on a sheet pan and cook in the oven for 9 minutes. Turn the package over halfway through. While the fish cooks, saute the broccoli rabe in a large saute pan with 2 tablespoons of olive oil until it is hot throughout.
Season with salt and pepper, to taste. To serve, put the pouches on each of four plates and open them at the table. Garnish with basil sprigs and pass the broccoli rabe.