Preheat the oven to 425 degrees F. Heat the olive oil and butter in a large skillet over medium-high heat. Add the garlic and cook until softened, about 1 minute. Add the greens and 1/2 teaspoon salt; cook, stirring occasionally, until the greens wilt and most of the liquid evaporates, 5 to 7 minutes.
Meanwhile, roll out the puff pastry on a lightly floured surface until large enough to fit in a 9-inch pie plate. Ease into the pie plate and trim the edge; crimp if desired. Scatter the goat cheese and Parmesan in the crust, then top with the sauteed greens.
Whisk the eggs, heavy cream, nutmeg, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl; pour over the greens. Bake until the crust is golden brown and the quiche is set, about 20 minutes. Slice into wedges.