For the shells: Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain pasta.
Warm the olive oil in a large, heavy skillet over medium heat. Add the pancetta and cook until lightly golden, about 5 minutes. Remove the pancetta from the pan with a slotted spoon and transfer to a large bowl. Add the spinach, ricotta cheese, asiago cheese, pepper, and nutmeg. Stir to combine. Stuff the shells with about 2 tablespoons of the spinach mixture each and place the stuffed shells in a large, buttered baking dish.
For the sauce: Melt the butter in a medium saucepan. Add the garlic and cook for 1 minute. Add the cream and bring to a simmer. Turn the heat to very low and add the 2 cups asiago cheese, parsley, and pepper. Stir until the cheese is dissolved. Pour the sauce over the shells. Top with the remaining 1/4 cup asiago cheese. Bake until golden on top, about 25 minutes. Remove from the oven and serve immediately.