Place mushrooms in a small saucepan with enough water to cover and simmer over low heat until tender (1-2 hours, topping up with extra water if necessary). Drain (discard water), cool, thinly slice and set aside.
Loosen noodles in a colander under cold running water to separate and set aside.
Process tofu in a food processor or blender until smooth then transfer to a squeezy bottle or piping bag without the nozzle and refrigerate until required.
Heat a wok over high heat, add vegetable oil and when it starts smoking, add mince and stir until starting to caramelise (2-3 minutes). Add chilli paste, ginger and shiitake, stir to combine (1 minute), then deglaze with sake (1 minute). Add stock, vinegar, soy sauce and sesame oil, bring to the boil and cook until a thick sauce consistency (1-2 minutes). Add udon and heat through, tossing continuously to coat well and prevent sticking to the wok (1 minute). Serve with tofu piped over in dots, and with sesame seeds and spring onions scattered on top.