Heat 2 tablespoons of olive oil in a large sauté pan over medium-high heat. Add 1 pound of ground beef and cook until browned, about 2-3 minutes per side.

In the same pan, add 1 diced onion and 1 diced carrot with a pinch of salt. Sauté over medium heat for about 5 minutes until softened.

Stir in 3 minced garlic cloves and 2 tablespoons of tomato paste. Cook for 5 minutes, constantly mixing.

Return the ground beef to the pan, add 1 cup of beef stock, 1/2 cup of beer, 1 tablespoon of Worcestershire sauce, and 1 teaspoon of thyme. Bring to a boil, then reduce heat and simmer for 35 minutes.

Mix 1 tablespoon of cornstarch with reserved beer to make a slurry. Stir into the pan and cook for another 5 minutes to thicken the mixture.

Preheat the oven to 400º F (200º C). Roll out the puff pastry to 1/8-inch thickness, cut 12 rounds, and place in a greased muffin tin.

Fill each pastry with about 1/3 cup of beef filling, add cheese on top, and cut 12 smaller rounds for the tops.

Seal each pie with the smaller rounds, brush with a beaten egg, and refrigerate for at least 15 minutes before baking for 18-20 minutes.