Heat 1 tablespoon butter in a skillet over medium heat. Add diced onion, carrots, and celery and cook for 5-7 minutes until softened. Add diced chicken and cook 3-4 minutes until no longer pink.
1 lb chicken breast, 1 medium onion, 2 medium carrots, 2 stalks celery, 3 tbsp butter
Push chicken and vegetables to the side of the pan. Add remaining 2 tablespoons butter to empty space, sprinkle flour over melted butter, and whisk for 2 minutes to cook roux. Gradually pour in milk while whisking until the sauce thickens, about 2-3 minutes. Season with salt and black pepper. Let filling cool for 10 minutes before assembly.
3 tbsp butter, 0.75 cup whole milk, 3 tbsp all-purpose flour, salt, black pepper
Roll out thawed puff pastry sheets on a floured surface. Cut each sheet into 4 squares, making 8 total. Place 2-3 tablespoons cooled filling on one half of each square leaving a ½-inch border.
1 package puff pastry
Fold pastry over filling to form triangles. Press edges with a fork to seal tightly. Brush tops with beaten egg wash, avoiding seams. Chill assembled pasties in the fridge for 10 minutes.
1 large egg
Preheat air fryer to 375°F. Place pasties in a single layer with 1-inch gaps between them. Cook for 12-15 minutes until golden brown and puffed, checking at 10 minutes to adjust time if needed. Let rest 5 minutes before serving.