Bring a large pot of water to a boil for the pasta. Bring a few inches of water to a boil in another pot for the broccoli rabe, salt the water, add the broccoli rabe. Cook until crisp-tender, about 3 minutes; drain. Chop.
In a large skillet, heat the oil, three turns of the pan, over medium-high. Add the sausage and cook, stirring and breaking up with a spoon, until browned, 5 to 6 minutes. Add the broccoli rabe, onion, chiles, garlic, lemon zest and oregano. Cook the sauce, stirring often, until the onion softens, about 3 minutes.
Salt the boiling water for the pasta, add the pasta and cook until al dente. Drain, reserving about 1/2 to 3/4 cup starchy cooking water. Add the starchy water to the sauce; remove from the heat. Toss in the pasta, cheese, mint and lemon juice; season.