Preheat oven to 375° F. In a non-stick pan, add 1 tbsp olive oil over medium high heat until it gets hot.
Add mushrooms and cook for ~ 2-3 minutes, until they start to get soft. Next add onions and cook for an additional 1-2 minutes. Then add carrot, garlic, celery, thyme, salt, and pepper. Cook for ~3-4 minutes. Set aside.
In food processor, add Field Roast sausage links and pulse ~8-10 times until sausages are the consistency of small pebbles. Add to large bowl. Repeat with walnuts and add to bowl with ground Field Roast. Repeat with sautéed vegetables and add to same bowl.
Add flour, breadcrumbs, and tomato paste to bowl and using a large wooden spoon or your hands, incorporate all ingredients and shape into a thick log, approximately 10 inches long and 4 inches across.
Place the log onto the thawed sheet of puff pastry. Wrap the log completely with the puff pastry by pinching the seams, cutting off any extra (which you can save for future use).
Using a very sharp knife, create slits in the top of the puff pastry. Mix together 1 tbsp extra virgin olive oil with 1 tbsp non-dairy milk. Brush the top of the puff pastry with mixture.
Place in the oven and cook for approximately 34-40 minutes, until puff pastry is golden brown. Serve with your favorite gravy.