Prepare the Potatoes: Place the cut potatoes in a large pot and cover with cold water. Bring to a boil and cook until the potatoes are tender (about 15-20 minutes).
Drain the potatoes and return them to the pot. Add the warm heavy whipping cream and sea salt.
Mash the potatoes until smooth, then fold in the shredded parmesan cheese and lightly beaten egg. Set aside.
Make the Pie Filling: In a large skillet, heat the olive oil over medium-high heat. Add the ground beef, salt and pepper. Cook until browned, breaking the meat into small pieces.
Add the chopped onion and minced garlic to the skillet. Cook until the onions are soft and translucent.
Sprinkle the all-purpose flour over the meat and cook for 2 minutes, stirring continuously.
Add the beef broth to the pan and scrape the bottom of the skillet to loosen any browned bits.
Stir in the tomato paste and Worcestershire sauce. Bring to a simmer and cook until slightly thickened (about 5 minutes).
Stir in the frozen vegetables and cook for another 5 minutes, or until the vegetables are heated through. Adjust seasoning with more salt and pepper, if needed.
Assemble the Pie: Preheat your oven to 375°F.
Transfer the meat and vegetable mixture to a 9x13 inch baking dish.
Spread the mashed potato mixture evenly over the top of the meat filling.
Brush the top of the mashed potatoes with melted butter.
Bake in the preheated oven for 25-30 minutes, or until the top is golden brown and the edges are bubbling. (Optional: Sprinkle on 1 cup of shredded cheddar cheese about 10 minutes into baking)
Garnish and Serve: Let the Shepherd's Pie cool for a few minutes before serving.
Garnish with chopped parsley or chives.