Arrange 2 racks to divide the oven into thirds and heat the oven to 425°F.

Empty 1 can whole tomatoes and their juices into a large Dutch oven, then use your hands to gently break apart and crush the whole tomatoes. Prepare the following, adding them to the pot as you go: Halve and thinly slice 1 small yellow onion. Crush 4 garlic cloves. Cut 2 tablespoons of the unsalted butter into small pieces. Place 4 sprigs thyme on top. Drizzle with 2 tablespoons olive oil and season with ½ teaspoon kosher salt, ¼ teaspoon black pepper, and ¼ teaspoon red pepper flakes.

Roast on the lower rack uncovered, stirring halfway through, until the garlic is soft and the mixture is bubbling and noticeably thickened, 40 to 45 minutes total.

When the tomatoes are ready, place the pot over medium heat. Stir the mixture to bring the thyme sprigs to the surface, then discard the sprigs. Add 1 box low-sodium vegetable broth and simmer, stirring occasionally, until the flavors have melded, about 10 minutes.

Stir in 1 tablespoon balsamic vinegar. Purée the soup in batches in a stand blender or with an immersion blender directly in the pot until smooth. Taste and season with more salt and pepper as needed.

Melt the remaining 2 tablespoons unsalted butter in a large bowl. Add ½ teaspoon Italian seasoning and stir to combine. Tear 8 ounces sourdough bread into rough 1-inch pieces directly into the bowl. Use your hands to toss and coat the bread in the butter. Transfer to a baking sheet and spread into an even layer. Bake on the upper oven rack, tossing halfway through, until crisp and golden-brown at the edges, 13 to 15 minutes total.

If needed, move the upper rack to the middle of the oven so the Dutch oven fits under the broiler. Heat the broiler to HIGH. Sprinkle half the cheddar cheese over the soup. Top with the croutons, arranging them in a single layer so they cover the soup. Sprinkle with the remaining cheddar and all of the Parmesan. Broil on the upper oven rack until the cheese is melted, 1 to 2 minutes. Garnish with fresh thyme leaves.