Make the sage-chile butter: Melt butter in a saucepan over medium heat. Add sage and red-pepper flakes. Simmer until sage is crisp, about 3 minutes.
Make the eggs and greens: Heat a large, heavy skillet (preferably cast iron) over high heat. Swirl in oil. Cook mushrooms with 1/2 teaspoon salt until golden and tender, 4 to 5 minutes. Reduce heat to medium. Stir in garlic, then greens and water. Cook, stirring, until greens wilt. Add unsalted butter, and stir until melted.
Push greens to make 4 wells. Crack 1 egg into each. Season with salt. Cook for 4 minutes. Let stand until whites are set but yolks are still runny, about 4 minutes. Drizzle with sage-chile butter. Garnish with cheese.
Very delicious and very handy for busy weeknights. We are going to have grits on the side the next time we enjoy this dish.