Combine the basil leaves, walnuts or nut of choice, and garlic in the bowl of a food processor and pulse until finely chopped. Add the Parmesan cheese, pepper, and salt, and pulse again to combine.
With the processor running, slowly pour the olive oil through the feed tube until the pesto is thoroughly blended. Add the lemon juice and pulse again to combine. Set the pesto aside.
Place the baby new potatoes in a large saucepan or pot and cover them with cold water by about 2 inches. Add a generous pinch of salt (this is important to season the potatoes).
Bring the water to a boil over high heat, then reduce the heat to medium and let the potatoes simmer until they are fork-tender about 15-20 minutes. Be careful not to overcook them; they should be tender but still hold their shape.
Once the potatoes are cooked, drain them in a colander and let them cool for a few minutes until they are cool enough to handle.
Cut the cooled potatoes in half (or quarters if they are larger) and place them in a large mixing bowl.
Add the pesto a little at a time, gently tossing until the potatoes are well coated. You may not need all of the pesto; use your judgment based on your preference for how "saucy" you want your salad.
The Pesto Potato Salad can be served warm or chilled. If not serving immediately, cover and refrigerate. Before serving, you might want to toss the salad again as the pesto can settle at the bottom.