Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a large nonstick frying pan, melt the vegan butter over medium-low heat. Add the potatoes and cook, stirring, until softened, about five minutes. Add the carrots and celery and cook, stirring occasionally, until the vegetables are soft, about ten minutes. Remove from the heat and scrape the mixture into a medium bowl.
In the same frying pan, melt the vegan butter over medium heat. Add the onions and cook until soft and translucent, about four minutes, stirring occasionally. Add the garlic and mushrooms and continue to cook, stirring occasionally, until the mush- rooms soften, three to five minutes. Add the concentrated vegetarian stock, red wine, thyme, and sage. Cook until the wine is reduced by half.
Add the mushroom mixture to the potato mixture. Stir in the panko crumbs and season with salt and pepper. (The filling can be made one day ahead. Keep in the fridge, covered.)
Arrange one sheet of puff pastry on a work surface with the long side facing you. Spoon half of the vegetable mixture lengthwise across the lower-half of the pastry, leaving a five to eight centimeter border, pressing the mixture together with your hands to form a roll. Fold the sides of the pastry over the filling, then roll it up, starting at the end closest to you. Repeat with the second Wellington.
Place the Wellingtons on the prepared baking sheet, leaving space between them. Lightly brush the Wellingtons with the melted vegan butter. Bake until golden brown, 25 to 30 minutes.
Transfer the Wellingtons to a long plate or a platter and garnish with the fresh thyme sprigs. To serve, cut into thick slices using a large, sharp knife.