Peel and dice 2 medium carrots and 1 medium yellow onion. Mince 4 cloves garlic.
Place 2 cups dried red lentils (about 1 pound) in a fine-mesh strainer. Pick through the lentils, discarding any broken or discolored ones. Rinse under cool water until the water runs clear, about 1 minute. Set aside to drain while you cook the vegetables.
Heat 3 tablespoons olive oil in a large pot or Dutch oven over medium-high heat until shimmering. Add the carrots and onion and sauté until softened, about 5 minutes. Add the garlic, 1 tablespoon curry powder, 1 teaspoon ground cumin, 1 teaspoon dried thyme, and 1/2 teaspoon kosher salt. Cook until fragrant, about 1 minute.
Add the lentils, 1 (15-ounce) can tomato sauce, and 6 cups broth, stir to combine, and bring to a simmer. Cover and reduce the heat to maintain a gentle simmer. Cook, stirring occasionally until the lentils are tender, 18 to 22 minutes.
Remove from the heat and stir in 5 cups spinach until wilted. Serve immediately.