Preheat your oven to 400°F (200°C) and ensure your puff pastry is thawing according to the package directions.

Heat the olive oil in a large skillet or Dutch oven over medium-high heat. Season the diced lamb evenly with salt and black pepper, then sear for 5-7 minutes until beautifully browned on all sides. Remove the lamb from the pan and set it aside on a plate.

In the same skillet, add the chopped onion and carrots. Sauté for 4-5 minutes until the onions soften and become translucent. Stir in the minced garlic and cook for an additional 30 seconds until fragrant.

Sprinkle the all-purpose flour over the sautéed vegetables and stir constantly for 1 minute to cook out the raw flour taste. Gradually pour in the beef broth, using a wooden spoon to scrape up any flavorful browned bits from the bottom of the pan.

Return the browned lamb to the skillet and stir in the diced potatoes, fresh rosemary, and Worcestershire sauce. Bring the liquid to a gentle simmer, cover, and reduce the heat to low. Let it cook for 20 minutes until the potatoes are tender and the gravy has nicely thickened.

Transfer the rich lamb and potato filling to a 9-inch deep-dish pie plate. Unroll the thawed puff pastry sheet and drape it carefully over the top of the pie plate, trimming any excess pastry and pressing the edges down to seal.

Brush the top of the pastry evenly with the beaten egg to ensure a shiny finish. Cut three small slits in the center of the crust to allow steam to escape while baking.

Bake in the preheated oven for 25-30 minutes, or until the pastry is puffed up and deeply golden brown. Let the pie rest for 10 minutes before slicing, and serve hot with a simple side of steamed peas or green beans.