Preheat the oven to 400°F (200°C). Wrap the beets in aluminum foil and roast for about 50 minutes.Transfer to a plate and reduce the temperature to 350°F (175°C).
When the beets are cool enough to handle, trim off their ends and peel. Set aside.
In a skillet, heat the vegetable oil and saute the leek for about 5 minutes, or until softened. Add the carrots and cook for 4-5 minutes more. Transfer to a plate.
In the same skillet, add the mushrooms and cook until they start to brown, for about 5 minutes. Salt and pepper to taste.
Line two baking sheets with parchment paper. On a floured surface, cut the puff pastry sheet in half and, working with one sheet at a time, carefully roll it out to form a square. Transfer the pastry to the baking sheets.
Divide the fillings into two parts and, working about 1 inch from the bottom edge, arrange the leek and carrot mixture. Top with the mushrooms. Make a “X” shaped cut into the center of each beet and stuff them with the goat cheese. Arrange 4 beets on each sheet. If they won't fit, carefully stretch the pastry sheet a bit.
Fold the unfilled side of the pastry over the filling to enclose it, gently sealing around the edges. Brush the top of the pastry with an egg.
Bake for 35-40 minutes, or until crust is golden brown. Enjoy!