Preheat oven to 350°.
Heat oil and garlic over medium heat in a cast iron skillet. Season chicken with salt and pepper, add to skillet.
Brown chicken until golden brown on each side, about 5 minutes each.
Remove chicken from skillet and add to a prepared 11x7 baking dish.
In a separate skillet, melt 2 T. butter and saute broccoli florets until slightly tender. Add to baking dish with chicken.
Meanwhile, in the hot chicken skillet, add 1/2 c. chicken broth to the pan to deglaze. Bring to a boil for 2-3 minutes.
Add 1/2 c. heavy whipping cream and return to a boil. Let reduce slightly.
Pour the sauce over the chicken and broccoli.
Sprinkle cheeses on top and add the final 1/2 tsp of alpine seasoning.
Bake at 350° for 30-40 minutes, depending on the thickness of your chicken. It's always a good idea to use a meat thermometer to check doneness before serving chicken.