Preheat the oven to 375 degrees . In a skillet, combine the beef, carrots, onion, celery, garlic and potato and cook, breaking up the meat, over medium-high heat until the beef is cooked through and some liquid has evaporated, about 15 minutes. Lower the heat to medium, add the beer and cook for 10 minutes. Add the cheese, 2 teaspoons salt and 1/2 teaspoon pepper. Let cool.
On a floured surface, roll 1 puff pastry sheet into a 12-inch round. Place in a 10-inch skillet or deep pie dish. Roll the remaining puff pastry sheet into an 11-inch round. Using a slotted spoon, transfer the meat mixture to the skillet. Lightly brush the edge of the crust with some of the egg wash, then place the second sheet on top. Press the edges to seal, then trim to a 1/2-inch edge.
Place the pie on a baking sheet. Brush the top with more of the egg wash and cut an "X" into the pastry. Sprinkle with 1 teaspoon salt and 1/2 teaspoon pepper. Bake for 45 minutes, tenting with foil if the crust darkens too quickly. Transfer to a rack to cool for 15 minutes.