Preheat the oven to 350ºF. Spread softened butter in each ramekin. Set aside.
Heat olive oil in a skillet, add mushrooms and sauté until tender. Add fresh thyme, and season with salt and black pepper.
Place the sautéed mushrooms to form a thin, even layer on the bottom of each ramekin. Gently slide 2 eggs into each ramekin. Drizzle 2 tablespoons of cream around each egg, then season with salt, black pepper and grated nutmeg. Top each egg with a light layer of grated gruyere cheese.
Place ramekins in a baking dish and transfer to oven. Carefully pour enough boiling water into baking dish to submerge ramekins ¾ of the way. Bake eggs en cocotte until whites are just set and yolks are still runny, about 20 minutes (12 minutes covered and 10 minutes uncovered). Very carefully remove baking dish from oven.
Finish with paprika butter and garnish with fresh chives or parsley. Serve with buttery toast. Enjoy!
Paprika Butter:
Melt remaining butter in a small skillet and sprinkle in the paprika. Cook for 1-2 minutes until fragrant.