Combine first 5 ingredients, stirring with a whisk.
Combine crawfish, All That Jazz Seasoning, and sour cream.
Heat a small nonstick skillet coated with cooking spray over medium-high heat. Add mushrooms and ham; sauté 3 minutes. Pour egg mixture into pan; let egg mixture set slightly. Tilt pan and carefully lift edges of omelet with a spatula; allow uncooked portion to flow underneath cooked portion. Cook 3 minutes; flip omelet. Spoon crawfish mixture onto half of omelet. Carefully loosen omelet with a spatula; fold in half. Gently slide omelet onto a plate; top with cheese. Cut omelet in half.