Preheat the oven to 350 degrees F. Butter a 2-quart baking dish and set aside.
Put the oil and garlic cloves in a large, high-sided saute pan over medium heat and cook the garlic until lightly golden brown on both sides. Add the bread cubes, a pinch of the crushed red pepper and some salt and pepper, and cook until the bread cubes are lightly golden brown on all sides. Using a slotted spoon, transfer the bread and garlic to the prepared baking dish.
In the same saute pan, add the tomatoes, sugar and another pinch crushed red pepper to then pan. Cook, stirring often, until soft and saucy, 10 to 15 minutes. Season with salt and pepper, add the basil, and then pour over the bread.
Lightly beat the eggs with some salt and pepper. Pour the beaten eggs on top of the tomatoes. Sprinkle the cheese on top. Bake until a paring knife inserted near the center comes out clean, about 25 minutes.
Let the strata rest about 10 minutes before serving.