Prepare the Rösti: Place the grated potatoes in a clean kitchen towel or sieve and squeeze out excess moisture. Transfer to a mixing bowl, add 1 tbsp of whole wheat flour, salt, and pepper. Mix well.
Start the Ragout Base: Heat 2 tbsp of rapeseed oil in a large pan or braising pan over medium heat. Add the diced onions and carrots. Sauté for 5 minutes until softened. Add the minced garlic and cook for another minute until fragrant.
Cook the Mushrooms: Add the sliced brown champignons and King Oyster mushrooms to the pan. Increase heat slightly and cook for 5-7 minutes until the mushrooms release their moisture and begin to brown.
Simmer: Stir in the dried thyme and savory. Pour in the vegetable broth and bring to a simmer. Add the drained white beans. Reduce heat to low, cover, and let simmer for 10 minutes to allow flavors to meld.
Fry the Rösti: While the ragout simmers, heat 2 tbsp of rapeseed oil in a second large frying pan (or crepe pan) over medium-high heat. Divide the potato mixture into 4 equal portions. Flatten each into a patty shape in the pan. Fry for 5-7 minutes per side until golden brown and crispy. Keep warm in the oven if necessary.
Finish the Ragout: Stir the vegan sour cream and ground almonds into the mushroom and bean mixture. Heat gently for 2-3 minutes (do not boil vigorously after adding cream). Stir in the apple cider vinegar. Season with salt and pepper to taste.
Serve: Place one crispy potato Rösti on each plate. Ladle the creamy mushroom and bean ragout alongside or over the Rösti. Garnish with fresh spring onion rings.