In a large saucepan, melt the butter over medium-heat. Add the onion and carrot, and cook until the onion is softened, about 3 minutes.
Add the broccoli, chicken broth, thyme, garlic powder, salt and pepper. Bring to a boil over medium-high heat, reduce the heat to medium, and simmer uncovered until the broccoli is softened, about 8 minutes.
With a slotted spoon, remove 1 cup of vegetables from the soup, coarsely chop, and set aside.
Using an immersion blender (or blender), blend the remaining vegetables and broth in the saucepan.
Place the flour in a small bowl. Add the half & half, a little at a time, stirring until smooth to create a slurry.
Bring the blended vegetable mixture to boil over medium-high heat. Whisk in the slurry. Continue whisking until the soup is thick and bubbling, about 3 to 4 minutes. Let boil for one additional minute.
Remove the soup from heat, and stir in the cheddar and Parmesan cheeses along with the reserved chopped vegetables. Taste and season with additional salt and pepper, if desired.