Break the eggs into a large bowl. Add milk, salt, pepper, green onions and Parmesan cheese; whisk until mixture is very well combined. Set aside.
Heat butter and oil in a 10-inch, well seasoned, cast iron skillet or other oven safe skillet over medium heat until sizzling (recipe note). Add asparagus; cook and stir 2-3 minutes, until bright green and crisp-tender. (4 to 5 minutes for tender-crisp.) Swirl pan to coat all sides with oil.
Whisk egg mixture once more and pour over asparagus. Remove from heat and sprinkle with cheese.
Bake 20-25 minutes, or until the frittata has puffed and the center is set. Remove from oven and let stand 10 minutes before slicing into wedges. Serve.
Notes
If you prefer a thinner frittata, cook in a 12-inch cast iron skillet and slightly reduce baking time (3 to 5 minutes).