Place an oven rack in the center of the oven. Preheat the oven to 375 degrees F. Butter a 9-by-5-inch nonstick loaf pan. Line the bottom of the pan with parchment paper. Set aside.
Place the eggs and milk in a large bowl. Add the bread and stir to combine.
Heat the butter in a large, high-sided skillet over medium-high heat. Add the onion, 1 teaspoon of the salt and 1/2 teaspoon of the pepper and cook until soft, about 6 minutes. In batches, add the spinach and cook until wilted, about 15 minutes. Drain in a colander. Press gently on top of the spinach to release excess moisture. Set aside to cool slightly. Transfer the spinach mixture to a food processor and blend until smooth. Pour the spinach into the bowl with the bread mixture. Add 3/4 cup of the Parmesan, the remaining salt and pepper and the nutmeg. Stir until smooth.
Spread half of the mixture on the bottom of the prepared pan. Lay the hard boiled eggs, end-to-end, in a line along the center of the spinach mixture. Add the remaining spinach mixture on top and press gently. Sprinkle the remaining Parmesan on top and bake for 1 hour. Cool 15 minutes, then run a thin metal spatula around the edges of the pan to release the loaf. Invert the loaf onto a cutting board, remove the parchment paper, and cut into slices.