1 pound fresh or frozen gnocchi
1 large delicata squash (about 1 1/2 pounds)
Extra-virgin olive oil, for the squash and gnocchi
Freshly ground black pepper
6 tablespoons unsalted butter
1/2 cup coarsely chopped walnuts
8 large fresh sage leaves, sliced into thin ribbons
1/2 cup coarsely grated or shaved Parmesan
Bring a large pot of salted water to a boil and line a baking sheet with paper towels.
Cook the gnocchi in the boiling water until they float, then remove them with a slotted spoon to the lined baking sheet to dry. Empty the pot, rinse and set aside.
Preheat a 6-quart air fryer to 375 degrees F.
Trim the ends of the squash, cut in half lengthwise and scoop out the seeds. Slice into roughly 1/4-inch-thick half-moons. Toss the squash in a medium bowl with 1 tablespoon olive oil, 1/2 teaspoon salt and several grinds of pepper.
Put the squash in the air fryer basket (reserve the bowl) and cook, flipping halfway through, until golden brown in spots, about 15 minutes. Transfer to a large serving bowl.
Preheat the air fryer again to 375 degrees F.
Add the gnocchi to the reserved bowl and toss with 2 teaspoons olive oil and 1/2 teaspoon salt. Transfer the gnocchi to the air fryer basket and cook, flipping halfway through, until golden and crispy, about 10 minutes. Transfer the gnocchi to the serving bowl with the squash.
While the gnocchi cooks, place the reserved pot over medium heat and melt the butter (make sure the pot is completely dry; any remaining water will make the butter splatter). Watching closely, cook the butter, swirling occasionally, until it begins to turn light golden and smell nutty, 2 to 3 minutes. Add the walnuts, sage and a pinch of salt and continue to cook, stirring; the butter will become foamy and begin to deepen in color. Continue to brown the butter until it is deep golden, about 2 more minutes. Pour the brown butter sage sauce over the gnocchi and squash, gently toss to coat and top with the Parmesan.