Begin by peeling and chopping the large potatoes into equal-sized chunks. Place them in a large pot, cover with salted water, and bring to a boil. Cook the potatoes for 15-20 minutes until they are fork-tender but not mushy. Once done, drain them well and return to the pot.

Using a potato masher, thoroughly mash the boiled potatoes in the pot. Add in the butter and milk, mixing until you achieve a smooth, creamy texture. Stir in the shredded cheese, salt, and pepper to taste, blending everything together. Allow the filling to cool for a few minutes while you get the crust ready.

Preheat your oven to 375°F (190°C) to prepare for baking the Easy Irish Potato Pie. Roll out the prepared pie crust on a floured surface until it’s about an eighth of an inch thick. Carefully place the crust into a pie dish, pressing it gently against the sides and bottom. Trim any excess hanging over the edges.

Spoon the creamy mashed potato filling into the prepared pie crust, spreading it evenly with a spatula for a smooth finish. If you're using a second crust, place it over the filling, making sure to seal the edges well by crimping them together. Brush the top crust with the beaten egg for a beautifully golden finish.

Transfer the pie to the preheated oven and bake for 25-30 minutes. Watch for the crust to turn a lovely golden brown and for the filling to bubble slightly at the edges. Once baked, carefully remove the pie from the oven and let it cool for about 10 minutes before slicing to ensure the filling holds together beautifully.