Speedy Cherry Tomato Sauce

olive oil 4 garlic cloves, minced 4 cups cherry tomatoes, quartered (2 pints or 1 quart) 1 teaspoon kosher salt 1/2 teaspoon fresh ground black pepper 1/4 teaspoon sugar 1/4 cup lightly packed fresh basil leaves, torn into small pieces

fusilli, rotini, penne, or other short spiral or tubular pasta, cooked according to package directions (reserve a cup of pasta cooking water; see notes) 1 cup fresh mozzarella cheese, diced (about 8 ounces)