Melt 3 tablespoons butter in heavy large pot over medium-high heat. Add garlic; sauté chicken quickly until tenders are browned, about 5 minutes. Remove chicken and set aside.
Brown mushrooms in same skillet. When mushrooms are just about done add asparagus and stir fry for about 2 minutes.
Toss prosciutto in to brown.
Add broth, sherry, lemon peel, thyme, 1 teaspoon salt, and 1/2 teaspoon pepper. Reduce heat to medium-low; simmer uncovered 20 minutes to blend flavors.
Whisk cornstarch mixture to combine and add to skillet; stir until mixture thickens slightly, about 1 1/2 minutes. Season sauce to taste with more salt and pepper, if desired. Rough chop chicken and add it back to pan.
Transfer mixture to 8×8 -inch glass or ceramic baking dish. Sprinkle Parmesan over.
Cut stars out of puff pastry with a cookie cutter and put on top of chicken mixture. Brush stars with an egg wash. Bake at 400 until bubbly and pastry is golden, about 30 minutes. Serves 2-4