In a large pot over medium-high, melt 2 tablespoons butter with 2 tablespoons oil. Set aside 1½ cups torn bread for thickening the soup. Add the remaining bread, ½ teaspoon thyme, ¼ teaspoon each salt and black pepper, and the pepper flakes (if using) to the pot. Cook, stirring occasionally and lowering the heat as needed to prevent the seasonings from burning, until the bread is golden brown, 2 to 4 minutes. Transfer to a large plate, toss with the Parmesan and spread in a single layer. Set aside to cool and crisp.

In the same pot over medium, melt the remaining 2 tablespoons butter. Add the onion, garlic, ½ teaspoon salt and the remaining ¼ teaspoon thyme; cook, stirring occasionally, until the onion is translucent but not browned, 5 to 7 minutes. Stir in the tomatoes, the 1½ cups reserved bread and 4 cups water. Cook, uncovered and stirring occasionally, until the bread is completely soft and the broth is slightly thickened, about 10 minutes. Remove from the heat and cool for about 5 minutes.

Using a blender and working in batches so the jar is never more than half full, puree the tomato mixture, streaming in the remaining 3 tablespoons oil, until smooth. Return the soup to the pot. (Alternatively, if you own an immersion blender, puree the soup directly in the pot; when almost smooth, slowly stream in the remaining 3 tablespoons oil while blending.) Cook over medium-low, stirring often, until heated through, 2 to 5 minutes. Thin the soup with water, if needed, to reach the desired consistency.

Off heat, taste and season with salt and black pepper. Serve topped with the croutons and sprinkled with additional Parmesan.