Cut the top quarter off of each of the tomatoes and reserve for later. Scoop out the pulp from the insides of the tomatoes and add to a blender. Sprinkle the tomato shells with 1/2 teaspoon salt. Season the pulp with the basil, broth, oregano, garlic, 2 tablespoons olive oil and 1 teaspoon salt. Blend on high speed until smooth, then pour the puree into a medium bowl. (You should have about 2 1/4 cups.) Add the tomato puree to the pulp puree and stir to combine. (It should yield 2 3/4 cups. If it doesn't, add more puree.) Stir in the rice.
Preheat the oven to 375 degrees F.
Spread the potatoes on the bottom of a 9-by-13-inch baking dish. Drizzle with 2 tablespoons olive oil and the remaining 1/2 teaspoon salt. Toss well to combine. Nestle the tomatoes into the diced potatoes. Fill each tomato almost all the way to the top with the rice filling. Place the tops back on the tomatoes. Pour any remaining filling around the tomatoes and over the potatoes. Drizzle the whole dish with the remaining 2 tablespoons olive oil and bake until the tomatoes are blistered on the outside and the rice is cooked through, 1 hour and 15 minutes.