Preheat oven to 400°F. Grease a 12-cup muffin tin with cooking spray.
In a large skillet, melt butter over medium heat. Add onion, celery, and garlic and cook until softened, about 5 minutes.
Add diced chicken and cook until browned, about 5-7 minutes.
Add frozen mixed vegetables, diced potatoes, and dried thyme. Season with salt and pepper to taste. Cook for another 5 minutes.
Sprinkle flour over the chicken and vegetable mixture and stir to combine. Cook for 1-2 minutes.
Slowly pour in chicken broth and milk, stirring constantly. Bring to a simmer and cook until the mixture thickens, about 5-7 minutes.
Roll out puff pastry on a lightly floured surface. Cut out 12 circles that are slightly larger than the muffin cups.
Spoon the chicken and vegetable mixture into the muffin cups, filling each one about 3/4 full.
Place a puff pastry circle on top of each muffin cup, pressing the edges down to seal.
Brush the tops of the puff pastry with beaten egg.
Bake for 20-25 minutes, or until the puff pastry is golden brown and the filling is hot and bubbly.
Let cool for a few minutes before serving. Enjoy!