Preheat the oven to 425° F. Heat the olive oil in a large pot over medium heat. Add the onion and carrot and cook until fragrant, about 5 minutes. Stir in the chicken and season with salt and pepper. Add the butter, thyme, paprika, garlic powder, and cayenne, and cook until golden and toasted, 2-3 minutes. Add the broth and milk. Bring to a gentle boil over medium heat. Add the pasta and broccoli and season with salt and pepper. Stir frequently until the pasta is al dente and the broth has been absorbed, about 8-10 minutes. Stir in 1 cup of cheese. If needed, transfer everything to an oven-safe casserole dish. Scatter the remaining cheese over top of the casserole. Bake 20 minutes, until the cheese is melted and beginning to brown. Serve warm with additional thyme and fresh basil.