01 - Set oven to 400°F.
02 - Boil peeled, cubed potatoes in salted water until tender, about 15 to 20 minutes. Drain thoroughly.
03 - Mash potatoes with butter and milk until smooth. Season with salt and pepper. Incorporate egg yolk if desired and set aside.
04 - Heat olive oil in a large skillet over medium heat. Add onion, garlic, and carrots; cook until softened, approximately 5 minutes.
05 - Add ground beef, breaking it up with a spoon. Cook until fully browned, about 7 to 8 minutes.
06 - Stir in tomato paste, Worcestershire sauce, thyme, rosemary, salt, and pepper. Cook for 2 minutes to blend flavors.
07 - Sprinkle flour over the mixture; stir to combine and cook for 1 minute.
08 - Pour in beef broth, stirring to deglaze the pan. Simmer 5 minutes until mixture thickens.
09 - Fold in frozen peas and cook an additional 2 minutes.
10 - Transfer beef filling to a baking dish. Spread mashed potatoes evenly on top and create decorative fork patterns if desired.
11 - Bake in preheated oven for 20 to 25 minutes until the top is golden and filling bubbles.
12 - Allow to rest 10 minutes before serving.