In a large skillet, melt 1 tablespoon of the butter. Add the onion quarters and cook over moderately high heat, turning occasionally, until golden brown all over, about 15 minutes. Transfer the onion quarters to a plate and let cool. Season the onions with salt and pepper and keep warm.
Add the port to the skillet and simmer over high heat until reduced by half, about 3 minutes. Add the orange juice and cook until reduced by half, about 2 minutes. Stir in the finely chopped chicken liver, reduce the heat to low and simmer until just cooked, about 1 minute. Remove the skillet from the heat. Stir in the mustard, season the sauce with salt and pepper and keep warm.
Spread the flour on a large plate and season with paprika, salt and pepper. Dredge the remaining halved chicken livers in the seasoned flour, shaking off any excess.
Melt the remaining 2 tablespoons of butter in a large skillet. Add the livers and cook over moderately high heat, stirring occasionally, until pink in the center, about 5 minutes. Transfer to paper towels to drain.
Arrange the spinach on a serving platter. Top with the onion quarters and chicken livers, drizzle the sauce over the top and serve.