Preheat the oven to 425°F. Set a 9x13x2 baking dish on the counter.
Heat 1 tablespoon of the olive oil in a large, heavy soup pot over medium heat. Add the sliced mushrooms to the pot and let them sit for a full minute. Season the mushrooms with pepper and stir. Let the mushrooms sit another minute or so. Season the mushrooms with salt and stir continuously until the mushrooms are soft and moist. Transfer cooked mushrooms to a plate or bowl and set aside.
Pour the remaining olive oil into the pot. Add the onions, carrots, celery, and leeks to the pot and stir. Cook and stir until the onions are quite soft, about 5 minutes. Add the rosemary and garlic to the pot and cook for 30 seconds, or until fragrant. Add the lentils to the pot and stir to coat in the oil/vegetables. Pour the balsamic vinegar and tamari into the pot and scrape up any brown bits that have accumulated.
Pour 3 cups of the vegetable stock into the pot. Bring the lentils to a boil and then lower the heat to a simmer. Cover the pot and cook until lentils are just tender and there is only a tiny slick of liquid left in the pot, about 18 minutes. Season the lentils with salt and pepper.
Turn the heat off and stir in the cooked mushrooms and remaining cup of vegetable stock. Transfer the stew to the 9x13x2 baking dish. Pat the stew down into the dish. There should be just enough liquid to keep things ever-so-slightly fluid—like a thick, but still broth-y, stew.
Trim the sides off of the potatoes so that you have 2 relatively even rectangular prisms of potato. Then, using a mandolin, slice the potatoes into ⅛ inch-thick slices. Lay out the slices and blot them dry with paper towel.
Layer the potato slices on top of the lentil stew, covering the entire surface. Drizzle the top of the potatoes with olive oil, season with salt and pepper, and scatter some extra rosemary leaves.
Slide the potato-topped stew into the oven and roast until the potatoes are very lightly browned on the edges and tender, about 35-40 minutes. Then, set the oven to broil and let the potatoes brown even further for about a minute. Serve the stew hot with sections of the potatoes on top.