To make the mashed potatoes:

1. Place potatoes in a pot and cover with cold water. Add a large pinch of salt and bring to a boil. Boil potatoes for 15-20 minutes, or until completely tender. Drain.
2. Use a potato ricer, or pass potatoes through a strainer using a wooden spoon into a mixing bowl. If you prefer more rustic potatoes, add cooked potatoes to a mixing bowl and mash thoroughly with a potato masher.
3. Stir in butter, then add milk to potatoes until they reach desired consistency. Season to taste with salt and pepper, then stir in the egg yolk. Set aside until filling is ready.

To make the filling:

1. Preheat oven to 400°F
2. Heat oil in a large skillet over medium heat. Sauté onion and carrots for 7-8 minutes, or until they begin to soften.
3. Add ground beef to the carrots and onions, using a spatula to break up the ground beef and mix with the vegetables. Stir in mushrooms and garlic. Season with salt and pepper. Cook 7-8 minutes, or until the beef is browned and the mushrooms are cooked (you want the mushrooms to shrink in size and lose their moisture.) Drain off any excess grease.
4. Add butter, flour, and tomato paste to the filling mixture, and stir gently for 1 minute.
5. Stir in beef broth, stirring out any lumps. Add Worcestershire sauce, rosemary, thyme. Bring to a simmer and simmer uncovered for 5-6 minutes or until gravy has thickened.
6. When the filling has thickened, stir in the peas. Adjust seasonings to taste. If using sprigs of fresh herbs, remove them from the filling before adding the potato topping.

To make the pie:

1. Top the filling with mashed potatoes. If using cheddar cheese, sprinkle the cheese over the potatoes. (If you didn’t cook the filling in a large, oven-safe pan, then transfer the filling to a buttered casserole dish before topping with the mashed potatoes.)
2. Bake for 30 minutes. If you want a browner crust, broil for a few extra minutes.
3. Allow to cool for 10 minutes, then serve.