01 - Preheat the oven to 400°F (200°C).
02 - In a large skillet over medium heat, heat the olive oil. Add the finely chopped onion and diced carrots, sautéing for 5 minutes until softened. Stir in the minced garlic and cook for an additional minute until fragrant.
03 - Increase the heat to medium-high. Add the ground beef to the skillet and cook until thoroughly browned, breaking it up with a spoon. Drain any excess fat if necessary.
04 - Stir in the tomato paste and Worcestershire sauce, cooking for 2 minutes. Pour in the beef broth, then add the dried thyme, dried rosemary, bay leaf, salt, and pepper. Bring the mixture to a simmer, then reduce the heat to low and cook for 15 minutes, stirring occasionally, allowing the flavors to meld.
05 - Remove the bay leaf from the beef mixture. Stir in the frozen peas and cook for 2 minutes until heated through. Remove the skillet from the heat.
06 - Meanwhile, place the peeled and quartered potatoes in a large saucepan. Cover with salted water and bring to a boil. Cook until the potatoes are fork-tender, which usually takes about 15–18 minutes. Drain the water and return the cooked potatoes to the pot.
07 - Add the unsalted butter and whole milk to the hot potatoes. Mash until smooth and creamy. Season with salt and white pepper to taste. If desired for a golden finish, mix in the egg yolk.
08 - Spoon the beef filling into a large baking dish (approximately 9x13 inches). Carefully spread the mashed potatoes evenly over the top of the beef mixture. Use a fork to create decorative ridges on the potato topping.
09 - Bake the shepherd's pie for 25–30 minutes, or until the potato topping is lightly golden brown and the filling is bubbling around the edges.
10 - Allow the shepherd's pie to rest for 10 minutes before serving. This helps the filling to set and makes for easier portioning.