Position a rack in the upper third of the oven; preheat to 450 degrees F. Cut the halibut into 4 pieces; arrange on a foil-lined baking sheet, cut-side down. Squeeze 1 lemon half over the fish and season with salt and pepper; spread the mayonnaise over the tops. Bake until the fish is almost cooked through, 6 to 7 minutes. Switch the oven to broil. Broil the fish until the tops are golden, about 3 minutes.
Meanwhile, combine the chives, dill and parsley in a small bowl; set aside. Toss the arugula with the juice of the remaining lemon half and 1 tablespoon olive oil; season with salt and pepper.
Top each fish fillet with the herb mixture. Drizzle the arugula salad and tomatoes with the remaining 1 tablespoon olive oil; season with salt and pepper. Serve with lemon wedges.