Bring a large pot of water to a boil for the pasta.
In a large skillet, heat the oil, two turns of the pan, over medium-high. Add the sausage and cook, stirring occasionally and breaking up with a spoon, until browned, about 5 minutes. Add the onion and garlic, season with salt and pepper and cook, stirring often, until the vegetables soften, 2 to 3 minutes. Add the stock; simmer until the liquid is reduced by about half, about 5 minutes. Season.
Salt the boiling water, add the pasta and cook to almost al dente. Reserve 1/2 cup of the starchy cooking water, then drain.
Add the peas to the sausage mixture, then add the escarole and cook, stirring often, until wilted, about 2 minutes.
Toss the pasta with the sausage mixture, over medium, adding some of the reserved starchy cooking water if too dry. Stir in the parsley, mint and lemon juice; season. Serve on a platter or in shallow bowls topped with a little grated cheese and drizzled with EVOO.