Step 1

Heat the oven to 450 F with a rack in the middle position. On a rimmed baking sheet, toss together the bread and 1 tablespoon oil, then distribute in an even layer. Bake until the bread is light golden brown, 5 to 7 minutes, stirring once halfway through.

Step 2

Meanwhile, in a large saucepan over medium, heat the remaining 2 tablespoons oil until shimmering. Add the onion, carrot, thyme and 1/2 teaspoon salt; cook, stirring occasionally, until the vegetables are softened and lightly browned, 6 to 7 minutes. Stir in the tomatoes with juices and 1/2 teaspoon each salt and pepper. Bring to a simmer over medium-high, then reduce to medium, cover partially and simmer, stirring occasionally, for 10 minutes.

Step 3

Stir in the broth and bring to a simmer, uncovered, over medium-high. Reduce to medium, then stir in the spinach and cook until just wilted, about 1 minute.

Step 4

Off heat, taste and season with salt and pepper. Divide the toasted bread among individual bowls and ladle in the soup. Serve sprinkled with the Parmesan and drizzled with additional oil.