Preheat air fryer to 375°. In a large skillet, cook beef, mushrooms and onion over medium heat until meat is no longer pink and vegetables are tender, 8-10 minutes. Add garlic; cook 1 minute longer. Drain. Stir in Worcestershire sauce and seasonings. Remove from the heat; set aside.
On a lightly floured surface, roll puff pastry into a 15×13-in. rectangle. Cut into four 7-1/2×6-1/2-in. rectangles. Place about 2 tablespoons potatoes over each rectangle; spread to within 1 in. of edges. Top each with 3/4 cup beef mixture; sprinkle with 1/4 cup cheese.
Beat egg and water; brush some over pastry edges. Bring opposite corners of pastry over each bundle; pinch seams to seal. Brush with remaining egg mixture. In batches, place pastries in a single layer in air-fryer basket; cook until golden brown, 10-12 minutes. Freeze option: Freeze unbaked pastries on a parchment-lined baking sheet until firm. Transfer to an airtight container; return to freezer. To use, cook frozen pastries as directed until golden brown and heated through, increasing time to 15-20 minutes.