Preheat your oven to 425-F.
Sautee the mushrooms in the butter or olive oil in a large skillet or pot until they've given off their liquid and are lightly browned, about 5 minutes.
Deglaze the pan with the red wine and reduce until most of the wine has evaporated.
Add all but 1/4 cup of the beef broth, the tomato paste, cubed potatoes, carrots, and salt and pepper to taste, and simmer in the broth until the veggies are fork tender. About 8-10 minutes.
Mix the reserved beef broth with the cornstarch to create a slurry.
When the veggies are barely fork-tender, add the cornstarch slurry to the pot and whisk to combine. Simmer until just thickened, 2-3 minutes.
Add the beef and peas and heat through.
Transfer the pot pie mixture to individual ramekins or oven-safe bowls, or to a pie dish.
Roll the puff pastry out on a floured surface until it is smoothed out and about 1/8th inch thick.
Cut circles larger than the top of your ramekins or pie plate so that about 1/2 an inch of pie dough overhangs the top.
Lay the puff pastry over the top of the ramekins and cut a few slits into the top to allow the steam to escape.
Whip the egg and brush the top of the puff pastry with the egg to create a glossy finish.
Transfer the ramekins to a baking sheet for baking in case they bubble over. Bake the pot pies at 425-F for 20 minutes or until the tops are golden brown.
Serve and enjoy!