Preheat the oven to 425°F and position a rack in the center.
Heat 1 tablespoon extra-virgin olive oil in a large, heavy-bottomed pot or large ovenproof skillet over medium heat. Add the chopped yellow onion and chopped carrots and cook, stirring occasionally, until softened and fragrant, about 5 minutes.
Add the cubed chicken breasts and season with kosher salt and black pepper. Cook, stirring occasionally, until the chicken is browned and no longer pink in the center, about 5–7 minutes.
Add 2 tablespoons salted butter, 2 tablespoons fresh thyme leaves, 1 teaspoon paprika, 1 teaspoon garlic powder, and 1/2 teaspoon cayenne pepper (or less, to taste). Cook, stirring, until the butter melts and the spices are fragrant, about 2–3 minutes.
Pour in 2 cups low-sodium chicken broth and 1 1/2 cups milk. Bring the mixture to a gentle boil over medium heat.
Add 1 pound short-cut pasta and 3 cups broccoli florets to the pot. Season with additional salt and pepper if needed. Stir frequently and cook until the pasta is al dente and most of the liquid is absorbed, about 8–10 minutes.
Remove the pot from the heat and stir in 1/3 cup grated Parmesan cheese and 1 cup of the shredded cheddar cheese until evenly combined.
If your pot is not oven-safe, transfer the mixture to an oven-safe casserole dish. Scatter the remaining 1/2 cup shredded cheddar evenly over the top.
Bake in the preheated oven until the cheese is melted and beginning to brown, about 20 minutes. Let the casserole rest a few minutes before serving.