Cook the pasta until al dente according to package instructions. Drain, place in a large bowl, and keep warm.
While the pasta is cooking, melt one tbsp of butter and brown the chicken cubes until fully cooked, about 4 minutes. Season with ½ tsp salt and pepper. Add to the bowl with the pasta.
Steam the broccoli according to package instructions in the microwave until tender-crisp. Add to the bowl with the pasta and chicken.
In the same skillet you browned the chicken, melt the remaining tbsp of butter, saute the minced garlic for 30 seconds, and add the flour. Cook until slightly golden, about 30 seconds.
Whisk in the milk, remaining ½ tsp salt, pepper to taste, and dried oregano until no lumps remain. Simmer until thickened, about 2-3 minutes.
Remove from heat and stir in the Parmesan cheese until melted. Pour the sauce over the chicken, pasta, and broccoli, and stir until fully combined.
Serve with extra Parmesan cheese on top.