In a small bowl, whisk egg and water until well blended. Lightly dust work surface with flour. Unfold puff pastry and dock dough with a fork. Using a 4-inch ring, cut 16 circles. Set 4 circles on a nonstick baking sheet. These are the base of the pot pies. Discard scraps. Using a 3-inch metal ring, cut circles in the center of the 12 remaining circles. Remove 4 of the 3-inch circles to the prepared baking sheet. These are the tops. Brush the outer edge of the 4-inch circles with the egg wash. Lay 1 of the 12 rings on top. Brush egg wash on that ring and place another ring on top. Repeat 1 more time until you have 3 rings stacked on top of each other. Repeat with remaining circles until you have 4 assembled. Brush the edges and tops with egg wash and bake in the middle of the oven until golden, about 20 minutes.
Meanwhile, season the shrimp with cayenne pepper, salt and black pepper. In a medium saucepan over medium-high heat, add oil and 1 tablespoon butter. Saute the shrimp, 1 minute per side. Remove shrimp and set aside. Add remaining butter to the pan and saute carrots, celery, mushrooms, and thyme, until softened about 10 minutes. Add wine, bring to a simmer and cook until most of the wine has evaporated. Add the cream and bring back to a simmer until the sauce thickens to a gravy consistency, about 12 to 15 minutes. Place the shrimp back into the sauce to coat. Remove thyme and season with salt and white pepper. Ladle the shrimp mixture into the pastry puff bowls and top with the smaller puff circles.