Boil the potatoes for the topping: Peel the potatoes and cut into 1-inch chunks. Place in a medium saucepan. Add enough cool water to cover the potatoes by 1 inch. Cover the pot and bring to boil over high heat. As soon as the water is boiling, uncover, then reduce the heat to a simmer. Cook, adjusting the heat as needed to maintain a gentle, steady simmer, until potatoes are fork tender, about 12 to 16 minutes. Drain the potatoes and return them to the saucepan. If the potatoes are very moist, heat the saucepan over low, shaking it occasionally, until any liquid on the surface of the potatoes has cooked off, about 1 minute, then remove from the heat.
Mix the topping together: With a potato masher or wooden spoon, mash the potatoes well. Scatter the butter pieces over the top. Add the Parmesan, salt, and pepper.
With a wooden spoon or rubber spatula, stir a few times, until the butter is mostly melted.
In a small bowl or liquid measuring cup with a spout, whisk together the milk and egg yolk.
Add to the potatoes. Stir to combine. The potatoes should be rich and fluffy.
Make the filling: Coat a 9×9, 8×10, or 11×7 casserole dish with nonstick spray. Place rack in the center of your oven and preheat to 400℉. In a large skillet, heat the oil over medium-high heat. Once it is hot and shimmering, add the onion and cook until it is beginning to brown, about 3 minutes.
Add the garlic and tomato paste. Stirring continuously, cook 30 seconds.
Add the beef (or lamb), salt, and pepper. Cook, breaking apart the meat, until it is browned and cooked through, about 4 minutes. If there is excess grease in the pan, carefully drain it off (this should not be a problem if your beef is around 93% lean).
Sprinkle the flour over the top, and cook, stirring continuously, for 1 minute.
Stir in the broth, Worcestershire, thyme, and bay leaf. With a rubber spatula or wooden spoon, scrape the bottom of the pan to remove any stuck-on browned bits. Bring the liquid to a boil, reduce the heat to low, then cover the pan and let simmer gently until the sauce thickens slightly, about 10 minutes. Remove the thyme and bay leaf. Taste and adjust the seasoning as desired.
Stir in the peas and carrots.
Transfer the meat mixture to the prepared dish, spreading it into an even layer.
Add the mashed potatoes by spoonfuls over the top, starting around the edges to create a “seal” (this will prevent the filling from bubbling up). With a spatula, gently spread the potatoes into an even layer.
Line a rimmed baking sheet with a piece of parchment paper, then place the dish on top (this will catch any parts of the filling that bubble over). Bake the shepherd’s pie uncovered on the center rack for 25 minutes, or until the potatoes begin to brown. Let cool for at least 10 minutes prior to serving.
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