Preheat the oven to 475 degrees . On a baking sheet, drizzle the squash with some of the EVOO; season with salt, pepper and nutmeg. Roast until just tender and brown at the edges, about 17 to 20 minutes.
Meanwhile, bring a large pot of water to a boil, salt it, add the penne and cook until al dente. Drain the pasta; reserve 1 cup of starchy cooking water.
While the pasta is cooking, heat 1/4 cup EVOO in a large skillet , four turns of the pan, over medium-high. Add the parsley stems, garlic, ginger, chile and scallion whites; swirl 2 to 3 minutes. Add the stock and reduce heat to simmer. Add half a ladleful of the pasta cooking water to the sauce. Add the pasta, parsley leaves, scallion greens, squash and cheese to the skillet. Stir to combine. Adjust the seasonings.