Preheat the oven to 425°F.

In an oven safe skillet over medium heat, melt the butter. Add the mushrooms, shallots and thyme and cook, stirring occasionally until the mushrooms brown, about 8 minutes.

Add the garlic and kale and cook until the kale wilts, about 2 more minutes.

In a large bowl, whisk together the eggs, milk, salt and pepper. Pour the eggs on top of the vegetables and cook until the edges of the eggs begin to set.

Sprinkle with parmesan cheese and transfer the skillet to the preheated oven. Cook until the eggs are no longer jiggly in the center and the edges are golden brown, 10-12 minutes.

Sprinkle more parmesan cheese and allow the frittata 5 minutes to rest before slicing and serving.