Melt butter in an oven-safe skillet over medium heat. Add mushrooms, shallots, and thyme, cooking until mushrooms are golden, about 8 minutes.
Stir in garlic and kale, cooking until kale is wilted, approximately 2 minutes.
Whisk eggs, milk, salt, and pepper in a bowl. Pour over the vegetables in the skillet, cooking until the eggs set at the edges.
Sprinkle Parmesan cheese over the frittata and bake in the oven until the center is set and edges are golden, 10-12 minutes.