Preheat the oven to 375°F (190°C).

In a large skillet, heat the olive oil over medium heat. Add the diced beef and cook until browned on all sides, about 5-7 minutes.

Add the chopped onion and minced garlic to the skillet and sauté until softened, about 5 minutes.

Stir in the tomato paste, beef broth, red wine, Worcestershire sauce, thyme, rosemary, salt, and black pepper. Bring the mixture to a simmer and cook for 10 minutes.

Add the sliced mushrooms to the skillet and cook for an additional 5 minutes.

Sprinkle the flour over the mixture and stir until it thickens, about 2 minutes.

Remove the skillet from the heat and transfer the mixture to a 9-inch (23 cm) deep-dish pie plate.

Roll out the puff pastry on a floured surface and place it on top of the pie plate, pressing down the edges to seal.

Brush the beaten egg over the top of the pastry.

Bake the pie for 35-40 minutes, or until the pastry is golden brown.

Allow the pie to cool for a few minutes before slicing and serving.