How to Make It
Preheat the oven to 325°. On a large rimmed baking sheet, toss the halved tomatoes with 2 tablespoons of the olive oil and season with salt and pepper. Bake the tomatoes cut side up for 1 1/2 hours, until soft and starting to brown. Sprinkle with the thyme leaves and bake for about 30 minutes longer, until the tomatoes are very tender and slightly shriveled but still juicy. Let cool.
Increase the oven temperature to 375°. Brush 16 bread slices with the remaining 2 tablespoons of olive oil; arrange 8 of the slices oiled side down on a large rimmed baking sheet. Top with the provolone and the unbrushed bread slices. Cover with the tomatoes, 4 cups of the Fontina and the remaining 8 bread slices, oiled side up. Press gently on the sandwiches and bake for about 15 minutes, until the bread is toasted and the cheese is melted.
Preheat the broiler. Toss the remaining Fontina with the Parmigiano-Reggiano and sprinkle on the sandwiches. Broil 3 inches from the heat for about 1 minute, until the cheese is melted. Transfer the sandwiches to plates and serve.