When boiling add the basil leaves and blanch for less than a minute. Drain. In a blender or food processor, place the basil, pine nuts, walnut, garlic, Parmiggiano, and olive oil. Blend on high.
When all the ingredients are well blended, add the heavy cream, and give it a quick pulse.
If using the potato, place the cut potato in a large pan of salted water and bring to a boil. Add the pasta and cook to al dente. Drain the pasta and reserve ¼ cup of the cooking water.
In a serving dish place the butter, the pesto, and add the cooked pasta. Add some of the water as you mix. The pesto should be of a nice light green color and the pasta should be moist. Generously grate Parmigiano cheese over the pasta.