Fill a large bowl with ice and cold water. Set aside. Bring a large pot of water to a boil. Quickly blanch the arugula in the boiling water, just for a second or two. Remove immediately with a small strainer and place directly in the ice bath to cool.
Once cool, scoop up the arugula and squeeze any excess water out of it. In a food processor, place the arugula, garlic, pine nuts, and olive oil. Blend until you have a creamy consistency. Place the pesto into a large bowl.
With a spoon or spatula, mix in the two cheeses and the softened butter. Season with salt to taste. Bring a large pot of water to a boil. Cook spaghetti according to instructions until al dente. Scoop cooked pasta directly into the bowl with pesto. Mix to combine. Serve immediately.