In a large bowl, combine the tomatoes, basil, garlic, mozzarella cheese, salt, pepper, and 1 cup of the olive oil. Mix well, cover, and let sit at room temperature for 2 hours.
After 2 hours, prepare the grilled cheese. Lay the slices of sourdough out on a work surface. Take about ¼ cup of the oil from the marinated tomato mixture and combine with the remaining olive oil in a small bowl. Brush each slide of the sourdough bread with the olive oil mixture.
Heat a nonstick sauté pan over medium-high heat and add 1 slice of bread, oil side down. Cover the bread with 2 slices of the mozzarella cheese, or enough to cover the bread, and ½ cup of the marinated tomato mixture; be careful not to let it fall off the bread.
Cover the mixture with 2 more slices of mozzarella, and then a slice of sourdough, non-oiled side down. Cook, covered, until one side of the bread is toasted and golden brown, about 5 minutes. Flip the grilled cheese and cook until the other side is toasted, about 2-3 more minutes. Repeat with the remaining slices of bread.