Rub the inside of a 3-quart baking dish with 1 tbsp. of the oil and set aside. Using a serrated knife, cut off the top third of each tomato and discard the tops. Cut 1⁄8" off the bottom of each tomato so that they'll sit upright in the baking dish; discard bottoms. Using a small spoon (a grapefruit spoon works the best), scoop out the seeds and pulp from each tomato and discard. Sprinkle the insides of each tomato with a little salt. Place the tomatoes upside down on a plate layered with paper towels and let them sit for 30 minutes to extract excess tomato juice, which may make the filling soggy.
Meanwhile, heat the oven to 400° and make the rice filling: Stir together the remaining olive oil, rice, parsley, feta, mint, and garlic and season with salt and pepper. (You should have about 4 cups of the rice mixture.) Arrange the tomatoes in the prepared baking dish, cut sides up, and fill each with about 2⁄3 cup of the rice mixture, mounding the tops slightly. Using a small brush, coat the tomatoes with some of the olive oil from the baking dish. Bake the tomatoes until the filling begins to bubble and brown lightly and the tomatoes soften, about 45 minutes. Serve hot or at room temperature.