Prepare the tomato soup: Remove cores from tomatoes and cut in half crosswise. Reserve until needed.

Heat butter and olive oil in a saucepan over medium-high heat. Add garlic and anchovy and cook until garlic is sizzling in the hot fat, 45 seconds to 1 minute. Carefully add tomatoes with their juices, salt, and red pepper flakes. Toss with a spoon every few minutes until tomatoes release their liquid and start to break down.

Once tomatoes have softened and started to collapse, reduce heat to medium-low and cook, stirring occasionally, until tomatoes have fallen apart and the soup resembles a coarse tomato sauce, about 45 minutes.

Meanwhile, prepare the cheese toast: Preheat the oven to 450 degrees F (230 degrees C). Line a baking sheet with foil or a silicone liner. Drizzle with 2 tablespoons olive oil.

Place bread slices on the prepared baking sheet. Top each slice with sliced basil, and then Cheddar cheese.

Bake in the preheated oven until the cheese is well browned and crispy around the edges, 15 to 20 minutes. Let cool to room temperature.

Remove soup from heat and pass through a mesh strainer into a bowl. Use the back of a ladle to push all the tomato mixture through. Discard the skins and seeds, which should be the only things left in the strainer.

Transfer the strained tomato soup back into the saucepan and place back over medium-low heat. Continue cooking to thicken slightly, if desired, or simply bring back to a simmer and serve immediately with the cheese toasts.