Toast the pine nuts in a small skillet over medium heat, stirring frequently, until they are golden brown and smell nutty, about 5 minutes.

Toast the unpeeled garlic cloves in the same skillet over medium heat until the skins are spotted brown and the cloves feel softened, about 8 minutes. Remove the skins once the garlic is cool enough to handle.

Process the toasted pine nuts, peeled garlic, basil, Parmesan, and a portion of the olive oil in a food processor until the mixture is smooth. Season with salt and pepper and set aside.

Bring 4 quarts of water to a boil in a large pot with 1 tablespoon of salt. Add the green beans and cook until they are tender-crisp and bright green, about 3 to 5 minutes. Use a slotted spoon to transfer them to a large bowl.

Add the potatoes to the same boiling water and cook until they are easily pierced with a fork, 10 to 12 minutes. Transfer them to the bowl with the green beans using the slotted spoon.

Add the gemelli to the water and cook until al dente. Reserve 1/2 cup of the pasta cooking water before draining the pasta into a colander.

Return the hot pasta, green beans, and potatoes to the pot. Add the pesto, butter, lemon juice, and remaining olive oil.

Toss the mixture together gently, adding the reserved pasta water as needed until the sauce is creamy and coats every piece of pasta. Serve immediately while hot.