Preheat oven to 325°. Grease an individual ceramic gratin dish with 1 tbsp. butter, leaving any excess butter in the dish.
Lightly sauté the Swiss chard stems in remaining butter, about 2 minutes, until softened. Add the chopped greens and sauté until wilted, another minute. (If using spinach, simply add all of the spinach at once and stir for 2 minutes, until wilted.) Season with nutmeg and garlic.
Distribute the chard around the prepared gratin, forming two small indentations in the center of the dish for the eggs (to prevent the yolks from resting against the sides of the pan).
Gently break the eggs into the wells.
Pour cream over top of the whole mixture, especially into the chard. Season lightly with salt and black pepper, and grate parmesan over top.
Cover the gratin with foil and bake for 18-20 minutes, until the whites are no longer jiggly.
Optional: drizzle with fresh lemon juice for a pop of acidity.
Serve warm with buttery bread, if desired.