In standing mixer with paddle attachment, or in a large bowl with rubber spatula, blend cream cheese and butter until creamy and smooth.
Stir in flour, Parmesan, sour cream, and 1/2 teaspoon salt until well blended. Turn out dough onto a clean surface and shape into a log 12 inches long and 1 1/2 inches wide.
Wrap tightly in plastic wrap and chill for at least 1 hour.
Meanwhile, heat oil in large skillet over medium-high heat until shimmering. Add shallot and sauté until softened and translucent, about 2 minutes.
Stir in mushrooms, 1/2 teaspoon salt and 1/4 teaspoon pepper. Sauté until mushrooms have released most of their liquid and it has begun to evaporate, about 15 minutes.
Stir in garlic until fragrant, about 30 seconds. Stir in sherry and 2 tablespoons cream. Cover and keep warm until serving time.
To bake crisps, preheat oven to 375 degrees. Remove dough from refrigerator and slice into 1/4-inch rounds, cleaning knife if it becomes sticky.
Arrange dough in single layer on ungreased baking sheets. Whisk together egg and remaining tablespoon cream. Brush on crisps. Sprinkle with salt. Bake until lightly golden brown but still soft, about 18 to 20 minutes.
Remove from oven. Top each crisp with 1 to 2 teaspoons mushrooms. Garnish with dill and serve