Peel potatoes and cut in 1-inch cubes.
Fill medium sized pot with water and bring to a boil. Add 2 tsp of kosher salt, add the potatoes and cook until tender, but al dente, about 20 minutes. Set aside to drain in colander.
Rinse basil leaves and pat dry; set aside.
To prepare the pesto, add basil leaves to a food processor, along with parsley, garlic, olive oil, walnuts and process by pulsing until a paste forms.. Add lemon juice and transfer mixture to a medium sized bowl; give it a stir. Add grated Pecorino cheese (a little at a time, tasting in between for taste and texture since you may not want to use all the cheese) and gently stir until well-blended. Taste and add more olive oil and or cheese until desired taste and consistency. Note: Flavors might be a little intense at first. If preferred, make a day ahead and refrigerate in air tight container to give flavors a chance to blend and mellow. Optional: For a less pungent garlic flavor, sauté the garlic in a little olive oil for about two minutes before processing the pesto. Fill a large, heavy-bottomed pot with water and bring to a boil over high heat. Add 1 tablespoon salt and the pasta and cook according to the directions on the package until al dente. Reserve one cup of the pasta cooking water in a glass measuring cup.
To a large serving bowl, add half the pesto mixture to the bottom of the bowl and about a cup of the chopped potatoes.
When the pasta is ready, add about 1/2 cup of the pasta water on top of the pesto and potatoes. Then add the pasta to the pesto and potato. Using tongs, gently toss together to combine, pulling upward until the pesto coats the pasta, and adding more of the reserved cooking liquid as needed. Add the remaining pesto and continue to gently toss.
Garnish with additional grated cheese and crushed red pepper if desired.
