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Bring a large pot of water to a boil. Meanwhile, put the nuts in a large skillet over medium heat and toast, shaking the pan occasionally, until fragrant and lightly browned, about 5 minutes.
When the water comes to a boil, salt it generously and add the potatoes. The potatoes should cook for 15 minutes total, so add the pasta to the water about 5 to 10 minutes later, depending on the package instructions, so the potatoes and pasta finish cooking at the same time. While the potatoes and pasta cook, put the basil, olive oil, nuts, cheese, garlic, and a large pinch of salt in a food processor or blender and process until smooth.
When the potatoes are fully tender and the pasta is al dente, scoop about a cup of their cooking liquid out of the pot, and then drain the potatoes and pasta. Toss the potatoes and pasta with the pesto, adding the reserved cooking liquid as needed to thin out the sauce. Taste and adjust the seasoning, and serve immediately, topped with additional grated Parmesan or pecorino Romano, if desired.