Heat oil in a saucepan over medium-high heat; add the sliced salami, cooking until golden, 3-4 minutes. Transfer to a paper towel to drain. In the same saucepan, cook the diced pumpkin and onion until soft and caramelized, 5 minutes. Stir in the garlic and spices, cook 1 minute. Stir in the milk and reduce to a gentle simmer. Cover and cook until pumpkin is tender, 12-15 minutes.
Transfer the sauce to a blender and purée until smooth. Coarsely chop the cooked salami, reserving 2 tablespoons for garnish and stir into the sauce with the basil and half the chives.
Cook the pasta according to the package directions and reserve 1 cup pasta water before draining. Toss the pasta with half the grated cheese and divide among 4 bowls. Thin out the sauce with spoonfuls of reserved pasta water, if needed, then ladle the sauce over the pasta and garnish with remaining salami, chives and grated cheese.