Lightly oil the grill grate. Preheat a gas grill to medium or prepare a medium-hot fire in a charcoal grill.
Cook the bacon in a large skillet over medium heat until lightly browned and most of the fat is rendered, 5 to 8 minutes, stirring often and adjusting the heat as necessary. Using a slotted spoon, transfer the bacon to a paper towel-lined plate to drain, reserving 1 1/2 tablespoons of bacon fat in the skillet.
Add the red onion to the skillet and cook over medium heat until it begins to soften, 2 - 3 minutes, stirring often. Add the garlic and cook until fragrant and the onion is softened, about 1 minute. Add the bread crumbs and basil and stir until well combined. Transfer the onion mixture to a medium-size bowl. Add the blue cheese and the drained bacon and set aside.
Cut about a 1/2-inch-thick slice off the stem end of each tomato. Using a paring knife, remove the cores of the tomatoes by cutting a cone-shaped piece about 2 inches deep and 1 1/2 inches wide in each. Cut a very thin sliver off the bottom of each tomato to help stabilize it as it grills.
Whisk together the olive oil and wine vinegar and drizzle about 1 teaspoon of the dressing over each tomato. Sprinkle the tops of the tomatoes lightly with salt and pepper to taste. Divide the bacon and blue cheese mixture evenly among the tomatoes, filling the cores and patting the mixture firmly in place to cover the top of each tomato.
Place the tomatoes on the oiled grill grate, cover the grill, and grill the tomatoes until they have softened and the stuffing is warmed through, 10 to 15 minutes. If you are using a charcoal grill, check the tomatoes after about 5 minutes and if the fire is too hot, causing the tomatoes to burn or cook too quickly, move them to the edge of the grill to continue cooking. If you need the center of the grill for cooking other foods, such as chicken or steak, the tomatoes can be cooked on the outer edge of the grill or on the grill’s top shelf over the main cooking grate; the cooking time will be longer.
Alternative Oven Method: Bake the tomatoes in a shallow oiled baking dish in an oven preheated to 400°F After baking the tomatoes for about 12 minutes, place them under the broiler for about 1 minute to lightly brown the stuffing.