Combine the chicken, miso, soy, sweetener, and baking soda in a large bowl and stir to coat evenly. Set aside to marinate for 30 minutes at room temperature, or up to overnight to tenderize.
While the chicken marinates, cook the mushrooms. Add the oil and mushrooms to a large nonstick pan and cook over medium-high heat until the mushrooms begin to release their liquid. Add the soy sauce, vinegar, and sweetener. Cook until no liquid remains, about 10-15 minutes total. Transfer to a bowl, sprinkle with furikiake and set aside. Do not clean the pan.
Season the cucumber with a generous pinch of salt and set in a strainer in the sink to drain for 15 minutes. Once drained, stir together all the salad ingredients, adjust seasoning to taste and set aside.
After the chicken has been tenderized, stir in the sake and vinegar and add it to the empty pan. Cook on medium-high heat until little to no liquid remains, about 15 minutes total, stirring often. Cook without stirring for the last few minutes to get a slight char on the pieces once the liquid is evaporated. Serve with the mushrooms, salad, and green onions sprinkled over top.