Heat the oil in a large saucepan over medium heat. Add shallots, garlic, and carrots, and cook for 5-7 minutes, or until shallots are soft.
Add the sliced mushrooms and cook for another 5 minutes. Add the diced Jerusalem artichoke, lentils, coconut milk, water, and salt. Bring to a boil and let simmer for about 25 minutes, or until lentils are cooked and Jerusalem artichokes are soft. Remove from heat and drain excess water. You want to drain as much water as possible.
Next, transfer to a food processor and pulse 5-8 seconds, you want to leave some whole lentils. Add the thyme, cumin, oats, nutritional yeast, maple syrup, and tomato paste. Pulse for another few seconds, until the oats are well incorporated and the mixture is soft but not too sticky. Test the mixture by taking a small amount between your hands, you should be able to form a patty. If it's too sticky, add more oats.
Transfer to a large mixing bowl, and stir in the dried cranberries, and cooked chestnuts. Mix well, slightly mashing the chestnuts.
Line a large plate with parchment paper. Using wet hands, form 4 large burgers (or 6 smaller ones) and transfer to the plate. Chill in the refrigerator for at least 1 hour. In the meantime, prepare the sauce.
Then, heat a tablespoon of oil in a large non-stick skillet over medium heat. Cook the burgers on each side for 3-4 minutes, or until golden brown.
Serve burgers as is, topped with the sauce, or with buns and your favorite toppings (salad, avocado, tomato, etc).
Sauce
Combine all the ingredients together in a small bowl. Taste and adjust seasonings if needed.