Cut the thick part of the chicken open with a knife so it will lie flat.
Season both sides of the chicken with salt and pepper.
Slice the onion thinly.
Remove the stems from the shimeji mushrooms and separate them into small clusters by hand.
Remove the stems from the oyster mushrooms and tear them apart by hand.
Remove the stems from the shiitake mushrooms and slice them.
Place the chicken skin-side down in a pan and turn the heat to medium-low.
Cook both sides while occasionally pressing down with a spatula. When fully cooked, set the chicken aside.
Add olive oil to the pan without removing the grease from the cooked chicken.
Add the shimeji, oyster, and shiitake mushrooms to the pan and fry slowly until browned. Do not stir until the mushrooms turn brown and shrink by about 40%.
Add a splash of sake to the mushrooms.
Add the sliced onions and fry further until softened.
In a bowl, mix flour, better than bouillon, and garlic with a small amount of milk, stirring well to avoid clumping.
Once the flour mixture is smooth, add the remaining milk and mix thoroughly.
Add butter to the frying pan with the onions and mushrooms.
When the onions are soft, add the white sauce and simmer until thickened.
Once the sauce is thickened, add the chicken back to the pan to heat.
Sprinkle with parsley, if you like, and it's done!