Boil the peeled and cubed russet potatoes for about 15 minutes until tender, then drain well and return them to the pot.
Mix the egg, chicken broth, milk, softened butter, salt, and pepper together, add to the potatoes, and mash until smooth and creamy, then set aside.
Preheat a large skillet over medium-high heat, spray with cooking spray, add the ground beef, season with salt and pepper, and cook for 4 to 5 minutes until browned.
Add the shredded carrot and chopped onion to the skillet and cook for an additional 5 minutes, stirring frequently.
In a small saucepan, whisk together the beef dry onion soup mix, dried onion, flour, and water, bring to a low boil, and simmer for 3 to 4 minutes.
Whisk the Worcestershire sauce into the gravy and cook until thickened.
Pour the gravy over the meat mixture and stir in the peas, corn, and frozen hash brown potatoes.
Simmer the meat and vegetable mixture for 5 minutes until heated through.
Preheat the oven to 375°F.
Transfer the meat mixture to a 13 x 9 inch casserole dish and spread evenly.
Spread the mashed potatoes evenly over the meat layer.
Sprinkle the sharp cheddar cheese evenly over the mashed potatoes.
Bake for 20 minutes until the cheese is melted and bubbly.
Let the shepherd’s pie rest for 5 to 7 minutes before serving.