See recipe notes for slow cooker method. Mix the salt, black pepper, oregano, and red pepper flakes in a small bowl. Set aside.
Heat the oil over medium heat in a deep skillet or large pot. Once the oil is shimmering, add the ground lamb and cook, breaking up the large chunks with a spatula, until browned, about 8 minutes. Add the onion, carrots, and celery and cook, stirring occasionally, until the vegetables are softened, about 10 minutes. Add the garlic and cook, stirring frequently, until fragrant, 1 to 2 minutes. Stir in the spice mixture and cook until fragrant, about 2 minutes. Add the tomato paste and cook, stirring occasionally, until caramelized, about 5 minutes. Pour in the broth and crushed tomatoes and mix well. Submerge the thyme sprigs in the sauce, cover, and reduce the heat to low. Simmer, stirring every 15 minutes, to prevent scorching, until thickened, about 45 minutes. Season with salt to taste.
Reduce the heat to low, discard the thyme sprigs, and stir in the coconut milk. Cover and simmer until the sauce is slightly thickened and warm, about 10 minutes.
Meanwhile, cook the dried pasta in boiling salted water until al dente according to package directions, about 12 minutes.
To serve, evenly divide the pasta among 4 shallow bowls. Top each with ½ cup of sauce and garnish with chopped parsley.
