1. In a small bowl, combine the minced shallot and vinegar and let stand for at least 5 minutes.
2. Whisk in 3 tablespoons olive oil to make a vinaigrette; season to taste with salt and pepper.
3. In a small skillet, heat 1/2 cup olive oil over medium-high heat. Add half the sliced shallots and cook only about 2 minutes, or until they are approaching golden brown. Using a slotted spoon, lift the shallots out of the oil and transfer them to a paper towel to drain. Season with salt. Repeat with the remaining sliced shallots. Set aside.
4. Arrange the tomatoes on a large platter. Season with salt and pepper. Just before serving, drizzle with the vinaigrette and top with the fried shallots and basil leaves.