Cook the rice according to package directions.
Add 1 tablespoon of butter to a large skillet (the same skillet you plan to make the gravy in) over medium-high heat. While the skillet heats, season the chicken breasts with a 1/2 teaspoon salt.
Pan sear the chicken for 4 minutes on each side over medium-high heat, then cover the skillet, turn the heat down to low, and let the chicken continue to cook for 10 minutes. Once cooked, transfer the chicken to a cutting board and let it rest for 10 minutes, then slice it into 1/2-inch pieces.
In the same pan that you used for the chicken, heat 3 tablespoons of butter over medium heat. Once melted, add the mushrooms, garlic, and onion and cook for 5-7 minutes, until the onions are translucent and the mushrooms are slightly browned.
Whisk the flour with a 1/2 cup of the broth. Then, pour it into the skillet, along with the rest of the broth and stir to combine. Let the gravy cook for 2-3 minutes, until the mixture boils and thickens, then stir in the heavy cream, salt, and pepper.
To assemble the casserole, add the brown rice to the bottom of a 9x13 casserole dish followed by the sliced chicken. Pour the gravy over the chicken, sprinkle the cheese on top, then top with the crushed crackers.
Bake at 350°F for 25 minutes, until the cheese is bubbling and browned.