Preheat the oven to 350 degrees F. Grease a 9-by-13-inch baking dish with 1 tablespoon softened butter.
Melt the remaining 3 tablespoons butter in a skillet over medium heat. Add the mustard greens, garlic and a pinch of salt and pepper and saute just until wilted, 2 to 3 minutes. Remove from the heat and cool slightly.
Whisk together the eggs, milk, baking mix, 1/2 teaspoon salt and some pepper in a bowl until well incorporated. Stir in the mustard greens, cheeses and hash browns. Pour into the greased baking dish. Bake until puffed lightly, golden brown and soft-set in the middle, 45 to 50 minutes. Cool slightly before slicing and serving warm.