In a large broiler-safe skillet, heat butter over medium-high heat. Add mushrooms; cook and stir 4-6 minutes or until browned. Add green onion; cook 1 minute longer. Remove from pan with a slotted spoon. Wipe skillet clean., For sauce, in a small saucepan, melt butter over medium heat. Stir in flour, salt and pepper until smooth; gradually whisk in milk and cream. Bring to a boil, stirring constantly; cook and stir 2-4 minutes or until thickened. Remove from heat; stir in cheese., Preheat broiler. For eggs, in a large bowl, whisk eggs, salt and pepper until blended. In same skillet, heat butter over medium heat. Pour in egg mixture; cook and stir just until eggs are thickened and no liquid egg remains. Remove from heat., Spoon half of the sauce over the eggs; top with mushrooms. Add remaining sauce; sprinkle with cheese. Broil 4-5 in. from heat 4-6 minutes or until top is lightly browned. Sprinkle with green onion.