Cook pasta in a large pot of salted water until almost al dente. Reserve 1 cup pasta water and drain remaining. Return cooked pasta back to pot and place over burner with heat turned off.
WHILE PASTA COOKS: Prepare pesto by combining green peas, parsley, mint, nuts, lemon juice, cheese, garlic, salt, and red pepper flakes in a food processor. Begin blending while gradually streaming in olive oil until mixture is smooth.
Immediately following adding cooked pasta back to pot, add butter and toss with pasta until melted. Add pesto and use tongs to toss and coat. Gradually stream in pasta water in 1/4 cup increments, tossing continuously, until pesto sauce clings to noodles and desired consistency is achieved (you may not need all the pasta water, but better to have extra for insurance). Remove from burner and add arugula; toss to combine. Transfer to a serving platter and garnish with additional pine nuts and cheese, if desired.