Place the potatoes in a saucepan along with 1 teaspoon of salt. Cover with cold water and bring to a boil over medium-high heat. Cook until the potatoes are tender when pierced with a fork, 10 to 12 minutes. Drain, return the potatoes to the pan and place it back on the heat just long enough to allow the water to evaporate. Transfer to a mixing bowl.
Add the butter, cream and milk to the potatoes and mash together. Season to taste with salt and white pepper, then add the egg yolk. Using an electric mixer on medium speed, whip the potatoes until light and fluffy, adding small amounts of milk as needed for a smooth consistency. Set aside.
Preheat the oven to 400°F and generously butter a 9-inch pie plate.
Heat the olive oil in a large pan over medium-high heat. Add the garlic and cook just until fragrant, 1 minute. Add the ground lamb and season with salt and pepper. Cook, breaking up the meat with a wooden spoon, until nicely browned, 6 to 8 minutes.
Carefully drain the fat from the pan, lower the heat to medium and add the carrots and thyme. Cover and cook until the carrots are crisp-tender, about 5 minutes.
Stir in the Guinness, raise the heat to medium-high and cook, stirring often, until about half of the liquid has evaporated, 2 to 3 minutes.
Add the tomato paste and leeks and cook until the leeks are tender and the mixture has thickened, 2 minutes longer.
Transfer the meat mixture to the prepared pie plate and spoon the potatoes on top as evenly as possible. Use a rubber spatula to smooth the potatoes to the edge and brush with melted butter.
Bake for 25 minutes or until the potatoes develop a pale golden crust. Allow to stand 5 minutes before serving.