In the bottom of a large skillet pan, melt the butter over medium heat.
Add in the onion, thyme, and 1/2 teaspoon salt. Cook, stirring occasionally, until the onion is soft (about 6 minutes).
Add in the rice and garlic and cook for 1 minute (stirring occasionally). Add in the broth, milk, and remaining salt. Add pepper to taste.
Bring to a boil, stirring occasionally to keep the rice from sticking to the pot.
Add in the cooked chicken, stir to mix it all together, and bring back to a boil.
Reduce the heat to medium-low, cover skillet with a lid, and simmer until the liquid is absorbed and the rice is tender* (about 15-18 minutes).
Remove skillet pan from heat. Stir in the frozen mixed vegetables and 1 cup of cheese.
Season with more salt and pepper if needed.
Top with the remaining cheese and place the lid on top of the skillet and let the cheese melt and the rice continue to steam (about 5 minutes).
Serve and enjoy!