Prepare the filling: Heat a large skillet over medium-high heat. Add the minced lamb and cook until browned. Remove the lamb from the skillet and set aside.
Sauté vegetables: In the same skillet, add the onion, carrot, and celery. Cook until softened, about 5 minutes.
Combine ingredients: Return the cooked lamb to the skillet. Stir in the Worcestershire sauce, thyme, salt, and pepper. Pour in the beef broth and bring to a simmer.
Thicken the sauce: In a small bowl, whisk together the flour and ¼ cup of cold water until smooth. Add the slurry to the skillet and cook, stirring constantly, until thickened.
Assemble the pie: Transfer the lamb mixture to a baking dish.
Make the mashed potatoes: In a large pot, bring the potatoes to a boil. Cook until tender, then drain. Mash the potatoes with the butter, milk, salt, and pepper.
Top the pie: Spread the mashed potatoes evenly over the lamb mixture.
Bake: Bake the pie in a preheated oven at 375°F (190°C) for 30-35 minutes, or until the top is golden brown and bubbly.
