Cook elbow noodles in salted water until slightly undercooked by 2 minutes. Place broccoli florets in the same pot and let it cook for 2 minutes. Drain water from noodles and broccoli. Set aside.
Preheat oven to 375°F (190°C). Lightly spray a 9 x 13 casserole dish with cooking spray.
In a big pan over medium heat, heat 1 tablespoon of butter. Add the onion and garlic and cook over low heat until the onion is tender, stirring occasionally (about 2-3 minutes). Add the flour and stir well, cooking an additional 2-3 minutes on medium-low heat.
Slowly whisk in the chicken broth until well combined over medium heat, then add the milk and bring to a boil. Simmer on medium-low heat, mixing occasionally until the sauce thickens (about 5 min).
Remove from heat and add 1 cup of grated Parmesan cheese. Stir until melted. Taste the sauce to decide if it needs salt.
In a large bowl, combine the shredded chicken, noodles, broccoli, corn, cheddar cheese, and the sauce. Mix well until evenly coated.
Pour the mixture into the prepared baking dish and top with 1/3 cup of grated Parmesan cheese. Bake for 15-20 minutes, or until cheese is thoroughly melted. Enjoy!