Make the Garlic Croutons: Preheat oven to 350°F. Toss together sourdough cubes, oil, garlic, salt, and pepper on a parchment paper-lined large rimmed baking sheet until evenly coated; spread in an even layer. Bake until lightly golden and crisp, 15 to 20 minutes. Let cool for 15 minutes. Set aside.

While Garlic Croutons are baking, make the Soup: Cut broccoli stems into ½-inch pieces to yield 1 ½ cups. Cut broccoli florets into 1-inch pieces to yield 3 cups. Set aside separately. Reserve any remaining broccoli florets and stems for another use. Heat oil in a large Dutch oven over medium. Add onion, carrot, shallot, garlic, and thyme; cook, stirring often, until onion is translucent, about 6 minutes. Stir 1 tablespoon broth with cornstarch in a small bowl. Add the cornstarch mixture and the remaining broth to the pot, and bring to a boil over medium-high. Add cauliflower florets and stems and reserved broccoli stems; return to a boil over medium-high. Reduce heat to medium; simmer, stirring occasionally, for 5 minutes. Add broccoli florets; simmer over medium, stirring occasionally, until vegetables are tender, about 5 minutes. Remove from heat.

Transfer 2 cups Soup (preferably mostly stems), cream, mustard, and 1 cup cooled Garlic Croutons to a blender. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening; process until smooth, 30 seconds to 1 minute. Stir purée back into remaining soup in Dutch oven; place over medium heat. Add cheese, ½ cup at a time, whisking constantly, until melted and creamy, about 2 minutes. Garnish with pepper and additional thyme. Serve hot with remaining Garlic Croutons.