Bring a large pot of salted water to a rolling boil.
Add the potato gnocchi and cook according to the package instructions until they float and are tender; reserve 1 cup of the cooking liquid, then drain the gnocchi in a colander.
While the gnocchi cooks, melt the butter in a large skillet over medium heat.
Add the drained gnocchi to the skillet and cook, stirring or letting them sit briefly, for about 1 minute to heat through and lightly brown.
Remove the skillet from the heat, stir in the pesto, and toss to coat the gnocchi; add reserved cooking liquid as needed to loosen the sauce.
Season to taste with kosher salt and freshly ground black pepper.
Spoon the gnocchi into shallow bowls, top with freshly grated Parmesan, and serve.
Optional leveled-up serving: top with baby arugula, toasted walnuts, and a squeeze of lemon juice if desired.
