Mix your flour, salt, and half the black pepper in a large bowl. Cut the cold butter and vegetable oil into small chunks, then work them into the flour with your fingertips until it looks like coarse breadcrumbs. Don't overwork it—you want pea-sized pieces throughout for that flaky texture.

Add ice water one tablespoon at a time, stirring gently with a fork until the dough just comes together. Stop before it gets sticky—this is key for a tender crust. Form it into a disk, wrap it in plastic, and chill for at least 15 minutes while you prep everything else.

Preheat your oven to 400°F. Heat the olive oil in a large skillet over medium heat, then add your diced carrots and potatoes. Cook for about 5 minutes until they start softening, stirring occasionally so nothing burns on the bottom.

Add the shredded chicken, frozen peas, chicken broth, heavy cream, and remaining black pepper to the skillet. Bring everything to a gentle simmer and let it cook for about 8 minutes until the sauce thickens up and the veggies are tender. Taste it and adjust seasonings as you go—trust me on this.

Pour the filling into a 9-inch pie dish, spreading it out evenly. Roll out your chilled dough on a floured surface until it's about 1/8 inch thick, then carefully transfer it over the filling. Trim any extra dough hanging over the edges and crimp them with a fork for that classic look.

Whisk together the egg with one tablespoon of water to make an egg wash, then brush it all over the crust. This is what gives you that beautiful golden brown color, so don't skip it. Cut a few small slits in the top so steam can escape while it bakes.

Bake for 35 to 45 minutes until the crust turns golden brown and you can see the filling bubbling slightly at the edges. Let it rest for 5 minutes before cutting—I know it's hard to wait, but it helps it stay together better. Serve it warm and watch everyone lose their minds over your chicken pot pie recipe.