Prep: thaw the puff pastry according to package instructions. Preheat the oven to 425F. Line a baking sheet with parchment paper and set aside.
Make the Filling: Melt the butter in a sauté pan over medium-high heat. Add the shallot and sauté for 3 to 5 minutes, until fragrant. Add the mushrooms, thyme, and black pepper and mix well. Sauté for 5-7 minutes, stirring occasionally. Add the garlic and tamari, then sauté for an additional 1 to 2 minutes. Remove from the heat and set aside.
Layer the Pastry: lightly dust a cutting board or clean work surface with flour, then place the puff pastry on top. Use a rolling pin to roll the puff pastry out to it is around 12” by 15-16” (it doesn’t need to be exact). Sprinkle the vegan mozzarella and parmesan over the surface of the pastry, leaving a 1” border on one long end of the pastry. Use a spatula to spread the cooked mushrooms over the cheese, maintaining the same border.
Roll: Use a brush or your fingers to lightly brush the clean edge of the pastry with water. Use two hands to roll the puff pastry up towards the edge, applying extra pressure once you get to the end to seal the pastry.
Cut: Note: I strongly recommend you use dental floss for this step; otherwise, place the rolled pastry in the fridge for 20-30 minutes first, so it’s firmer when you use a knife. Slide a long piece of floss under the puff pastry roll, then cross the two strands over the top to form a 1” piece. Continue to pull on the crossed strands until they cut through the roll entirely, then transfer to the baking sheet (see video for more details). Repeat with the remaining puff pastry until you have 15-16 pieces.
Bake: Bake in the top rack of the oven for 18 to 22 minutes, until the pastry is golden. Remove from the oven and serve warm or cold. Store leftovers at room temperature, in a container with a loose-fitting lid, for up to 3 days.