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. Serve warm or at room temperature.