In a large broiler-safe skillet, heat butter over medium-high heat. Add mushrooms; cook and stir until browned, 4-6 minutes. 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 until thickened, 2-4 minutes. 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.