Preheat and oil the grill or preheat the oven to 375°F.
Cut each tomato in half crosswise. Combine the cheese and basil in a small bowl. Spoon equal portions of the mixture on the cut side of each tomato half. Drizzle with melted butter and season with salt and pepper to taste.
Place the tomatoes, cut-side up, on the preheated grill. Cover and grill for 3 minutes. uncover and grill until the top is bubbling, another minute or so. Alternatively, place the tomatoes on a rimmed baking sheet in the preheated oven and bake until the top is bubbling, about 10 minutes (if you want the tops to brown, preheat the broiler and place the grilled tomatoes under the broiler for 30 seconds or so).
Remove from the grill or oven and serve.
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