- Preheat an oven to 450°F. Prepare a hot fire in a grill.
- Place the tomatoes on a baking sheet and toss with 1 tablespoon of the olive oil. Arrange them, cut sides up, on the sheet and season with salt. Roast until tender, about 20 minutes. Let cool, then cut in half crosswise.
- Meanwhile, bring a large pot three-fourths full of salted water to a boil over high heat. Add the pasta and cook until al dente (tender but firm to the bite), about 10 minutes. Drain, rinse under cold running water and drain again. Set aside.
- Brush both sides of the tuna fillets with 1 tablespoon of the oil. Season well with salt and pepper. Place on the grill rack 4 to 6 inches above the fire and grill until lightly browned, about 3 minutes. Turn and cook for 3 to 4 minutes more for medium, or until done to your liking. Transfer to a cutting board, let cool and cut into 3/4-inch cubes.
- In a food processor or blender, combine the basil leaves and the remaining 1/2 cup oil. Pulse or blend until chopped to a coarse puree. Add the vinegar and season with salt and pepper. Pulse or blend until combined.
- In a large bowl, combine the pasta, tomatoes and any accumulated juices, tuna, mozzarella, parsley and basil dressing. Toss gently and serve.