Place avocado oil in a wide saucepan over medium heat. Add diced onions and saute until slightly caramelized (about 15 minutes), adding water as necessary.
Add diced carrots to pan and saute until softened, about 3-5 minutes. Stir in garlic, thyme, sage, and rosemary, and saute for 1 minute or until fragrant.
Add diced mushrooms and coconut aminos to pan. Cook until the mushrooms have have released their moisture (6-8 minutes). Once the mushrooms have become relatively dry, stir in coconut aminos and cook for 1 minute.
Remove pan from heat and allow to cool for at least 10 minutes.
Pour walnuts into the bowl of a food processor or blender. Blend until the walnuts have become a rough powder - about 10 seconds.
Drain your can of chickpeas into a medium mixing bowl, saving the aquafaba. Do not rinse.
Mash the chickpeas into a rough paste using a potato masher or pastry blender. We are not trying to make hummus; you still want an abundance of chickpea texture once you are done.
Add ground walnuts, panko, 4 tablespoons of aquafaba, salt, and tomato paste to mashed chickpeas. Mix in cooled sauteed vegetables.
Stir until ingredients are thoroughly combined and are easy to manipulate. If mixture appears wet, add additional panko. Form mixture into a roughly 9-inch-long log.
Remove thawed puff pastry from refrigerator and, on a clean surface, roll it into a roughly 10” x 13” rectangle. Transfer to a greased or lined baking sheet and preheat oven to 400 degrees F.
Place the mushroom log in the middle of your puff pastry. Wrap the log like a gift, ensuring the edges are completely sealed. If you have extra dough along the edges, feel free to trim them off.
Flip the loaf so that it is seam-side down and cut a series of cross-hatches into the top of the loaf, to allow it to release excess moisture.
Brush the loaf with 2 tablespoons of aquafaba to add extra shine before placing the loaf in the oven. (Can substitute one egg beaten with one tablespoon of water if not vegan.)
Bake at 400 degrees F for 45-55 minutes, until center of loaf is hot and pastry is golden brown. Allow to rest for 10 minutes before slicing and serving.