Place the chicken on a flat surface, skin side down, and open flaps. Season with salt and pepper. Heat 2 teaspoons butter in a skillet over medium heat. Add shallots. Reshape breasts and place seam side down in skillet over shallots. Add wine, chicken brother, and season with more salt and pepper. Toss in the parsley, cover and simmer 15 to 20 minutes, until breasts are just cooked through. Remove from heat.
Meanwhile, prepare the spinach in a separate saucepan. Pour water in saucepan and add salt to taste and spinach. Stir until wilted. Bring to a boil, stirring occasionally, and cook about 5 minutes. Drain spinach, pressing out as much liquid as possible. Coarsely chop spinach and season with salt, pepper, and nutmeg. In another skillet heat 1 tablespoon butter until it begins to brown. Add spinach and stir until heated through. Set aside.
Drain the liquid from the chicken and reserve. Remove the chicken and cover, keeping warm.
In a saucepan heat the remaining 2 tablespoons of butter. Stir in the flour and whisk until blended. Add the reserved chicken cooking liquid, stirring with the whisk. After about 5 minutes is should be thickened and smooth. Add the heavy cream and cook 2 minutes. Remove from heat and add the egg yolk, stirring briskly.
Meanwhile preheat the broiler. Spoon the spinach down the middle of a glass baking dish. Arrange chicken pieces on top of spinach. Pour sauce over both the chicken and spinach. Sprinkle the cheese on top. Run under the broiler until the cheese is melting and bubbly.