Preheat the oven to 425°. Slice off the top 1/2 inch of each tomato and reserve the tops. Scoop out the tomato cores and seeds. Cut a very thin sliver off of the bottom of each tomato to help them stand up straight. Arrange the tomatoes in a 9-by-13-inch glass or ceramic baking dish.
In a bowl, combine the goat cheese with the egg, garlic, basil, salt, pepper, and 2 tablespoons of the olive oil. Spoon the goat cheese mixture into the tomatoes, mounding the filling 1/2 inch above the rim. Cover with the tomato tops and drizzle with the remaining 2 tablespoons of olive oil.
Bake the tomatoes for 35 minutes, until tender and browned in spots and the cheese is hot. Let stand for 15 minutes. Drizzle each with a bit of the balsamic. Serve warm or at room temperature.