Bring a large pot of salted water to a boil. Prepare the pasta according to the package instructions, cooking until slightly past al dente. Before draining the pasta, scoop up 3/4 cup of the starchy pasta water and set aside. Drain the pasta and toss it with a little olive oil so that it doesn’t stick together. Spread the pasta on a baking sheet and let it (and the reserved pasta water) cool to room temperature.

Make the pesto: In a food processor, combine the pine nuts, lemon juice, garlic, salt, and several grinds of pepper and pulse until well chopped. Add the basil and pulse to combine. With the food processor running, drizzle in the olive oil and pulse until combined. Add the cheese, if using, and briefly pulse to combine. Set aside.

In a large bowl, combine the pasta, pesto, 1/2 cup of the reserved pasta water, lemon juice, salt, and several grinds of pepper and toss. Fold in the tomatoes, mozzarella, and arugula and season to taste, adding more pasta water as needed to coat the salad in a lightly creamy sauce. Top with the pine nuts, basil, and red pepper flakes, if using. Serve at room temperature.