Preheat oven to 350 degrees.
Prepare frozen broccoli according to instructions on package.
Prepare rice according to instructions on package.
In large saucepan, combine condensed cream of chicken soup, condensed cream of mushroom soup with 1-1/4 cups of milk. Add 2 cups of shredded cheddar cheese. Turn on low heat. Stir constantly until cheese is melted.
Over cheese has melted, remove from heat and cover with a lid to keep warm.
In a large skillet over medium heat, melt 1 tablespoon of unsalted butter. Add chopped onions. Stir constantly until onions are translucent. About 2-3 minutes.
Pour translucent onions in a large mixing bowl. Add cooked broccoli, cooked rice (white or wild), cheese mixture, preferred amounts of salt and coarse ground black pepper.
Stir until well combined.
Pour into a 9x13 baking dish.
Cover top remaining 1-1/2 cups shredded cheddar cheese.
Bake in preheated oven for 30 to 35 minutes OR until dish is bubbly, and cheese is melted. Cooking time may vary based on oven.
Serve hot.