Bring a large pot of salted water to a boil. Cook the penne according to the package directions.
While the pasta is cooking, toast the walnuts in a dry skillet over medium heat until fragrant. Reserve a small handful for garnish and put the rest in a blender or food processor with the garlic, cheese, and vinegar. Pulse several times until finely ground. With the machine running, pour in 4 tablespoons of olive oil and blend until thick and smooth.
When the pasta is done, reserve about 2 cups of the starchy cooking water and drain the pasta. Pour the pasta back into the hot pan and drizzle with another tablespoon of olive oil. Toss to coat. Add the walnut sauce and some of the reserved pasta water. Toss to coat. Add more pasta water as necessary until all the pasta is coated. Season to taste with salt and pepper.
Add the arugula and pomegranate seeds and stir gently. Serve immediately with extra parmesan cheese.