Heat the oil in a large ovenproof sauté pan (see notes), over medium-high heat. Once hot, add the lamb (or beef) and cook - breaking apart with a wooden spoon - until browned, about 5 - 8 minutes. Drain off the excess fat, season with salt and pepper, then remove with a slotted spoon to a plate. Reserve.
Add more oil if needed, lower the heat to medium, then stir in the onion, sautéing until translucent, about 3 minutes. Add the carrots and celery and continue sautéing, until the veggies have softened, about 8 minutes. Stir in the garlic and herbs, cooking for another minute. Season with a pinch of salt and pepper.
Pour in the red wine to deglaze the pan, using a wooden spoon to scrape all the browned bits from the bottom of the pan. Cook until the wine is almost fully evaporated.
Stir in the tomato paste and stir to combine, cooking for a minute or two until it turns from bright red to a deeper red color. Once dissolved, stir in the flour and cook for 1 minute.
Pour in the beef stock and Worcestershire sauce. Cook until it begins to boil, then return the ground beef to the pan. Cover, lower the heat, and simmer for 45 minutes.
Add the peas and corn, mixing to combine. Cover and continue cooking for another 15 minutes.
Taste and adjust seasoning as needed. Reserve.
Preheat oven to 375ºF.
Place potatoes in a large pot of salted water. Bring to a boil over medium-high heat. Cook for 15 - 20 minutes, or until potatoes are fork-tender. Drain in a colander.
Using a ricer, rice the potatoes back into the pot. Stir in the butter (sliced), warm cream, 1 cup of the cheese, yolk, and a generous pinch of salt and pepper. Stir until the butter and cheese have melted and everything is well combined. Taste and adjust seasoning.
Top the meat with dollops of the mashed potatoes, using a spatula to spread them out evenly. Melt the remaining butter in the microwave, then drizzle over the mashed potatoes. Finish with the remaining shredded cheese.
Place the sauté pan on a baking sheet (to catch any drips) and in the preheated oven. Bake until the potatoes are lightly browned and the edges are bubbling, about 30 minutes. If desired, pop under the broiler for 3 to 5 minutes for deeper browning.
Let the shepherd's pie rest for 10-15 minutes before serving.