Preheat the oven to 200°F
In a large bowl, toss the tomatoes with 1/2 cup of the olive oil and salt and pepper to taste. Place cut side down on a small baking sheet and bake for about 2 hours, or until the tomatoes are softened.
Remove the tomatoes from the oven and let cool.
Transfer the cooled tomatoes to a colander and set aside to drain while you make the pesto.
Combine the garlic and Parmigiano in a blender and pulse until the garlic is roughly chopped. Add the basil and pulse 7 or 8 times, or until the leaves are shredded. With the blender running, slowly add the remaining 1 cup olive oil, blending until smooth.
Toast the pine nuts in an 8-inch sauté pan over medium heat, tossing frequently, until golden brown, 3 to 4 minutes. Transfer to a plate to cool.
To serve, arrange 3 tomato halves cut side down on each plate. Place a ball of mozzarella in the center and spoon 2 tablespoons of the pesto onto each ball of mozzarella. Sprinkle with the pine nuts and garnish with basil leaves.