Char the peppers on all sides over an open flame on a gas stovetop or under a hot broiler (with the oven door cracked to avoid steam buildup in the oven). Put the charred peppers in a bowl, cover, and let them rest until cool enough to handle, about 10 minutes.
Meanwhile, heat the extra-virgin olive oil in a soup pot. Add the garlic, carrot, celery, and onions to the pot and season with salt and pepper, to taste. Cook until the vegetables soften, about 10 minutes, then put them into a food processor with the diced tomatoes. Seed and peel the peppers and add them to the processor. Puree the mixture until smooth, then pour it into soup pot and stir in the stock.
Heat a panini press or a large heavy skillet. Make sandwiches using 2 slices of cheese and 2 slices of ripe tomato on 4 of the bread slices. Sprinkle with salt and pepper, to taste, and top with some torn basil. Cover with the remaining 4 slices of bread and add them to the panini press or hot skillet. If using a skillet, top with a second smaller pan and weight it down with heavy cans. Press the sandwiches to toast and melt the cheese, a few minutes on each side or 5 minutes in a hot press.
Serve the soup in bowls with the Panini alongside.