Melt butter in a Dutch oven with oil over high heat; swirl pan to blend. Add chopped mushrooms. Cook and stir over high heat about 2 minutes. Add onion and chicken-breast pieces. Reduce heat to medium-high and cook until chicken is just cooked through, stirring in garlic the last minute or so.
Add undiluted mushroom soup and the cheese spread (or if using Velveeta, dice it). Stir until cheese is melted. Remove pan from heat. Stir in the cooked rice, thawed broccoli, water chestnuts, sour cream and pepper. Spoon mixture into a 2-quart baking dish sprayed with a nonstick product.
Bake at 400 degrees, uncovered, for 15 to 20 minutes. Remove from oven. Top with Cheddar cheese and breadcrumbs (a bit of grated Parmesan mixed in with the crumbs is optional). Return to oven until cheese melts.
Recipes adapted from the USA Rice Council in Houston, Texas.