Place a baking sheet on the middle oven rack; preheat to 200˚ F. Place each chicken breast between 2 sheets of plastic wrap and pound with a meat mallet or heavy skillet until 1/4 inch thick. Season with salt and pepper.
Put 1 cup flour on a plate. Beat the eggs in shallow bowl. Heat a large cast-iron skillet over medium-high heat, then add the vegetable oil. Coat 2 chicken breasts in the flour, then dip in the beaten eggs. Carefully add to the skillet and cook until golden brown and cooked through, about 3 minutes per side (reduce the heat slightly if the chicken is browning too quickly). Remove the chicken to the hot pan in the oven. Repeat with the remaining 2 chicken breasts.
Discard the oil and wipe out the skillet. Add the butter and melt over medium-high heat. Whisk in the remaining 1/4 cup flour; cook, whisking, until the mixture is bubbling but not browning, 1 to 2 minutes. Gradually whisk in the milk, then the chicken broth until smooth. Bring to a boil, then reduce the heat and simmer until thickened, 4 to 5 minutes. Stir in the spinach and Parmesan and cook until heated through and well combined, 3 to 5 minutes. Remove from the heat and stir in the lemon zest and nutmeg; season with salt and pepper. Serve over the chicken.