In a small bowl, stir together 3 tablespoons of the olive oil, the garlic, 1/4 teaspoon of the salt and pepper to taste. Let stand 30 minutes. 

Meanwhile, bring a large pot of water to a boil. Add the tomatoes and cook 5 minutes.  Drain well.  Pass the tomatoes through a food mill and set aside. (Alternately, let the tomatoes cool slightly and slip off their skins. Place the peeled tomatoes in a food processor and process, using quick on/off motions, until the tomatoes are smooth.)

Heat the remaining 4 tablespoons olive oil in a large wide pot over medium heat. Add the onion and sauté until softened, about 5 minutes. Add the tomatoes, remaining 2 teaspoons salt and pepper to taste. Adjust the heat to simmering and cook, stirring occasionally, until the sauce is lightly reduced, about 20 minutes.

Place the mozzarella in the simmering sauce; only half of each ball should be submerged.  Remove from the heat and let stand for 10 minutes. Meanwhile, chop 8 of the basil leaves and cut the remaining 8 leaves crosswise into thin strips.

Remove the mozzarella from the sauce, being careful not to splash the tops of the mozzarella with sauce. Stir the chopped basil into the sauce and scrape it into a serving bowl.  Place the mozzarella back in the sauce.  Stir the basil strips into the garlic oil and spoon the mixture over the mozzarella.  Serve at once with toasted slices of good country‑style bread or bruschette.

Note: Bocconcini–”mouthfuls”, are small balls of fresh mozzarella that measure about 1 1/2-inches in diameter and weigh about 2 ounce each, although they can vary in size. If you cannot find them, cut larger balls of fresh mozzarella in half and place them cut side down in the sauce to poach them.