Bring a large pot of water to a boil and salt generously. Add the pasta and cook until al dente. Drain, reserving about 1 cup pasta water.
Roughly chop some of the walnuts and set aside. Pulse the remaining walnuts and garlic together in a food processor, then add the arugula, kale, basil, cheese, lemon juice, sea salt and pepper and pulse to puree. With the motor running, slowly add the oil and process until smooth. The pesto can be refrigerated in an airtight container for up to 1 week.
Add the pesto to a large bowl. Add the hot pasta and toss to coat, adding a splash of pasta water to thin, if necessary. Transfer to a serving bowl and garnish with more cheese, a drizzle of olive oil and a sprinkle of chopped walnuts.