Preheat oven to 300 F. In a large skillet over medium heat add olive oil and mushrooms. Saute until mushrooms are partially cooked ( about 5 min.). Add arugula, spinach and a pinch of salt and continue to saute until mushrooms are fully cooked and greens have wilted (approx. and additional 5 min.).
Transfer greens and mushrooms to an oven proof dish and make 4 indentations (like little wells) in the greens. Crack and place one egg in each indentation.. Cook in the oven until the white part of the egg is set (approx. 10-15 min.).
While the eggs are cooking, in a small bowl mix yogurt , garlic and a pinch of salt. Set aside (leaving at room temperature).
In a small saucepan, melt the butter and add the chili flakes ,paprika, and a pinch of salt. Cook until the butter starts to bubble and form a foam. Add the sage and continue cooking for a few more seconds. Set aside.
Remove eggs from the oven and spoon yogurt mixture over the center of the dish and drizzle the chili butter over the top of yogurt and eggs. Serve right away.