Adjust an oven rack to the middle position and heat the oven to 400 degrees F. Put the potatoes in a large saucepan, cover with water, and bring to a boil. Reduce heat to a simmer, then cook until the potatoes are fork-tender, 15 to 20 minutes.
Meanwhile, cook the lamb in a large ovenproof skillet over medium heat, breaking up the meat with a wooden spoon, until no longer pink and the fat is rendered, about 3 minutes. Drain the grease through a fine-mesh strainer set over a large bowl. Discard all but 1 tablespoon of the rendered fat.
Drain the potatoes well and return them to the saucepan over low heat. Mash the potatoes thoroughly with a potato masher. Fold in the butter until melted, then stir in the warm milk and egg yolks. Season with salt and pepper to taste. Cover pan and set aside while preparing the filling.
FOR THE FILLING:
Heat the 1 tablespoon of reserved lamb fat over medium heat, using the same skillet you cooked it in. Once the fat is shimmering, add the onion, carrots, and 1/2 teaspoon salt and pepper. Cook until the vegetables are softened, 5 to 7 minutes. Stir in the garlic and cook until fragrant, about 30 seconds. Stir in the flour and tomato paste and cook, stirring constantly, until the flour is incorporated, about 1 minute.
Slowly stir in the stock/broth, thyme, and Worcestershire sauce, scraping up the browned bits from the bottom of the pan. Bring mixture to a simmer, then reduce the heat to medium-low and cook until the sauce has thickened, 3 to 5 minutes.
Off the heat, stir in the drained lamb and frozen peas, season with salt and pepper to taste, and smooth the filling into an even layer. Dollop the potato topping evenly over the filling, then spread it into an even layer, covering the filling completely and anchoring the potatoes to the sides of the skillet. Bake the pie at 400 F. until the top is golden brown, 20 to 25 minutes. Let the shepherd's pie cool for 10 minutes before serving.