Add the mushrooms to a food processor and pulse until crumbly, 10-12 times. Be careful not to over-process.
Heat 2 tablespoons of olive oil in a medium pan over medium heat. Add the onion and sauté for 3-5 minutes. Add the vegetable broth and sauté until the onions are translucent and most of the broth has evaporated.
Add the garlic and sauté for 3 minutes more, or until fragrant.
Add the mushrooms and soy sauce and sauté until most of the liquid has released from the mushrooms and evaporated, 10-12 minutes.
Push the sautéed veggies to the sides of the pan and add a bit more olive oil to the center. Sauté the thyme in the oil until fragrant, then incorporate into the rest of mixture.
Add the spinach and sauté until wilted.
Grease an 18 x 13-inch baking sheet with nonstick spray. On a lightly floured surface, use a rolling pin to roll out puff pastry sheet to fit the baking sheet. Transfer the puff pastry to the pan.
In center third of the pastry, add a layer of potato slices. Top with half of the mushroom mixture and spread in an even layer, about ¾-inch thick. Add another layer of potatoes and the rest of the mushrooms, and top with a final layer of potatoes.
Fold one side of pastry over the filling, then the other side. Seal the top and bottom ends over the filling. Score the top of the Wellington diagonally with a sharp knife.
In a ramekin or small bowl, combine the non-dairy milk with the remaining tablespoon of olive oil. Brush over the Wellington. Decorate with more pastry, if desired. Poke air vents in the side of the Wellington.
Bake for 35-40 minutes. until golden brown and puffed.
Let rest for 10 minutes before cutting into thick slices.