Adjust an oven rack to the middle position and heat the oven to 475 degrees. Cook the lamb in a 12-inch ovensafe nonstick skillet over medium-high heat, breaking apart the meat, until no longer pink and the fat has rendered, 3 to 5 minutes. Transfer the lamb to a strainer set over a medium bowl and let drain, reserving 1 tablespoon of the fat.
Add the reserved tablespoon fat to the skillet and return to medium heat until shimmering. Add the onion, cinnamon, and nutmeg and cook, stirring often, until softened, 5 to 7 minutes. Stir in the garlic, tomato paste, and oregano and cook until fragrant, about 30 seconds.
Stir in the broth, 1/2 cup of the cream, and 1/2 teaspoon salt, then add the pasta and drained lamb. Increase the heat to high and cook at a vigorous simmer, stirring often, until the pasta is tender, 9 to 12 minutes.
Whisk the remaining 1/2 cup cream and cornstarch together in a small bowl, then stir into the skillet and continue to simmer until slightly thickened, about 1 minute. Off the heat, stir in 1/2 cup of the Pecorino and season with salt and pepper to taste. Sprinkle the remaining 1/2 cup Pecorino over the top and bake until lightly browned, 5 to 10 minutes. Serve.