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1
In a medium saucepan, heat 2 tablespoons of butter and the tomato paste over medium heat. Cook, stirring until the butter is foamy and the tomato paste begins to stick to the bottom of the pan, about 2 minutes. Stir in the tomatoes, thyme, sugar, half a garlic clove, and 4 cups water. Season generously with salt and pepper. Crumble 1 bread slice into pan, stirring to combine. Bring the soup to a boil, reduce the heat, and rapidly simmer for 10 minutes until the tomatoes have broken apart.
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2
Heat a large cast iron skillet over medium heat. While the soup simmers, make the sandwiches: Rub the inside of the bread slices with the cut side of the remaining garlic clove. Stack the cheese on top of two of the pieces of bread and sandwich it with the remaining bread slices (garlic-rubbed sides of all bread slices should be facing the cheese). Spread the exposed side of the bread slices with the remaining butter. Add the sandwiches to the hot pan and place a second, heavy skillet over the top. Fry the sandwiches, turning once, until the cheese is melted and the outside of the sandwiches are crispy, about 8 minutes.
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3
Remove the thyme stems from the cooked tomato soup. Working in two batches, transfer the tomato soup to a blender and puree until smooth. Season the soup with salt, pepper, and vinegar. Serve the soup in bowls, each accompanied with half of a sandwich.
