Bring a large pot of salted water to a boil. Cook the prosciutto in a large skillet over medium-high heat until it renders some fat and begins to brown, about 2 minutes. Add the garlic and thyme and cook until the garlic just begins to soften, about 30 seconds. Remove the skillet from the heat and add the wine. Return to the heat and cook until the wine evaporates slightly, about 2 more minutes. Add 1/4 teaspoon salt and pepper to taste. Add the cream and nutmeg and cook, stirring occasionally, until the sauce thickens slightly, about 3 more minutes.
Meanwhile, add the pasta to the boiling water and cook as the label directs, adding the asparagus during the last minute of cooking. Reserve 1/2 cup cooking water, then drain the pasta and asparagus.
Add the cheese to the cream sauce and stir to combine. Add the pasta and asparagus and toss to coat, adding some of the reserved cooking water to loosen, if needed. Season with salt and pepper. Divide among bowls and top with more cheese.