Preheat oven to 350 degrees.
In a large mixing bowl, crack 12 eggs. Add in milk, salt, and pepper. Whisk vigorously.
In a cast iron skillet, heat butter on stove top until it's melted.
Add sliced mushrooms and shallots. Saute for 2-3 minutes, and add in arugula and cook until wilted (about 1 minute).
Pour eggs over top. Crumble goat cheese across top and cook on stove top until it just starts to set.
Place in oven, and cook for 15 minutes. Remove and let sit for 5 minutes to firm up.