Cook asparagus in large pot of lightly salted, boiling water until tender-crisp, about 2 to 3 minutes. Drain into a colander and rinse under cold water; drain well.

Preheat broiler. In a large skillet with oven proof handle, heat butter. Add green onions and cook over medium heat until slightly softened, about 1 minute.

In a bowl, whisk eggs with mustard, salt, and pepper. Stir in Swiss cheese. Reduce heat to medium-low, pour egg mixture into skillet, and stir gently to combine. Scatter cooked asparagus over eggs. Cover and cook until eggs are almost set on top, 6 to 8 minutes.

Sprinkle Parmesan cheese on top and broil about 5 inches from heat, about 1 minute or until top is set and lightly flecked with brown. Cut into wedges and serve.