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Bring a large pot of salted water to a boil over a high flame. Meanwhile, make the pesto sauce by blending or processing 8 Tbsp. olive oil along with the garlic, arugula, walnuts and cheese until smooth. Season with salt and pepper to taste and set aside.
In a large sauté pan over a medium flame, heat the remaining oil until hot. Add the sausage and brown on all sides until cooked through, about 8-10 minutes. Remove to a cutting board. Add the pasta to the rapidly boiling water and cook until al dente, about 4-5 minutes. Strain well, reserving 1 cup cooking liquid. Return the pasta and cooking liquid to the large pot over a very low flame. Slice the sausage and add to the pasta along with the pesto sauce. Stir gently until combined, seasoning with salt and pepper to taste. Divide the pasta between 4 plates and serve with more grated cheese, if desired.