Toasting and grinding your own sesame seeds will substantially more flavorful than using pre-ground, and it only takes a couple of minutes. If you want to try this, do it before cooking the mushrooms. First put the sesame seeds in a frying pan and toast them over low heat. When a few sesame seeds start to pop, remove from the heat. Let them cool for a minute or two and then grind them in a mortar and pestle or spice grinder, leaving a few unground for texture. If you don't want to bother with this step, you can use pre-ground store bought sesame seeds.
Slice and prepare the mushrooms and mix the sauce ingredients in a small cup with a fork.
Heat the neutral flavored oil in a large, wide, nonstick skillet over medium-high heat.
Add the mushrooms to the hot pan with a pinch of salt. They should sizzle on contact. If the mushrooms are overcrowded and can't be spread out in a single layer, cook them in two batches. Sauté, stirring occasionally, for about 5 minutes, until well seared. If using any small, delicate mushrooms, add them toward the end of cooking.
When the mushrooms are browned to your liking, (about 5-10 more minutes), stir in the splash of sake to deglaze. Stir in the sauce and cover, steaming the mushrooms for another 2-3 minutes. Remove from heat, and then stir in the greens. The greens will wilt on contact with the hot mushrooms. Let everything sit for a minute to meld.
Divide the salad into two portions and serve warm.
Notes
Don't limit yourself to the mushrooms in the ingredients list, use whatever mushrooms you have. If you've got bags of mushroom odds and ends, use them! If you've never tried shijemi mushrooms, I highly recommend them! They are cheap and require zero chopping. The only mushroom I might not use would be chanterelles, because for some reason I don't associate those with asian flavors and prefer them in more Italian-type dishes.