Preheat oven to 350ºF. Arrange potato slices, shingle-style, in overlapping layers in the bottom of a 9-inch square casserole dish. Sprinkle with salt and pepper. Arrange carrot slices over potatoes and then scatter onions over carrots. Sprinkle with a little more salt and pepper.
Place shredded cabbage over vegetables; sprinkle with a little more salt. Press down gently. In a small bowl or cup, mix the vegetable broth and vinegar. Carefully pour over the cabbage and vegetables in the casserole dish. Sprinkle thyme over cabbage. Cover dish with foil and place on the top rack of the oven. Bake for about an hour.
Meanwhile, line a baking tray with parchment paper and place shiitake mushrooms on the tray in a pile. Drizzle with several tablespoons of olive oil and sprinkle liberally with salt. Toss together to coat mushrooms with salt and oil then arrange mushrooms in a single layer. Place the tray on the middle rack of the oven. Bake for about 20 to 25 minutes, until mushrooms look dark brown–almost burnt, and are very crisp. Remove tray from oven and let mushrooms cool and continue to crisp.
Meanwhile, cook sausages in a lightly oiled pan over medium heat until well-browned on all sides, about 8 to 10 minutes. Set aside.
Remove casserole from oven after an hour. Check potatoes and carrots; you should be able to pierce them easily with a fork, but still feel a little resistance. Arrange the browned sausages on top of the cabbage and re-cover with foil. Place dish back in the oven for another 20 minutes. This steaming of the sausage plumps them up and makes them moist and tender.
To make the mustard sauce, whisk together all ingredients in a small bowl.
Remove casserole from oven and lift foil. Check potatoes and carrots for doneness. When done, serve immediately by placing a heaping spoonful of vegetables on a plate, top with sausage, sprinkle with shiitake crisps, and drizzle with mustard sauce.