In a medium saucepan, melt 2 tablespoons butter with 1 tablespoon EVOO, 1 turn of the pan, over medium heat. Add the spaghetti and toast until deeply golden, about 4 minutes. Stir in the rice, then stir in 2 1/2 cups chicken broth; season with salt and pepper. Cover and bring to a boil; lower the heat and simmer for 15 minutes.
While the rice is working, using a food processor, process the spinach into a paste. Stir into the rice and let stand for 2 to 3 minutes.
Preheat the broiler. In a small skillet, heat the remaining 2 tablespoons butter over medium heat. Add the shallots, season with salt and pepper and cook until tender, 3 to 4 minutes. Whisk in the flour and cook for 1 minute, then whisk in the Worcestershire sauce, pomegranate juice and remaining 1/2 cup chicken broth. Cook until thickened, about 5 minutes.
Coat the lamb chops with the remaining 1 tablespoon EVOO and season with salt and pepper. Broil, turning once, until browned and medium-rare, about 5 minutes. Serve with the gravy and rice pilaf.