My husband came home with something called Korean a-uk. It's a green.

A bipimbap type preparation comes to mind. Chiffonade the leaves and combine with maybe spinach, shitake, firm tofu, soy sauce, sesame oil and so on. Finish with a fried or raw egg which you stir into the hot bowl.

Looks like Romaine and Spinach hooked up. I'd try it as a raw salad green first.

Well, that is a puzzler.
If it tastes good raw. Maybe just use it in a salad. With some carrot bits and a dressing of...lime and lemon juice, some toasted sesame seeds (ground to paste in a mortar)..a clove of garlic, a tsp or more of sugar, with light veggie oil and soy sauce and a touch of sesame seed oil. Whisked in a mini prep or blender--drizzling the oil at the end.
And topped with the lightly cooked enoki.

It looks like it would really cook down if heated unless you where making a Goma-ae type cold vegie side dish with sesame dressing.
For that basically the same dressing..but the spinach is just lightly blanched...by holding the bunch up by the end and swrilled in a pot of boiling salted water.

It sort of looks like a bloated maple leaf. It has a softer texture softer than spinach and a slightly grassy taste, to me anyhow.

What's the texture of the greens? I'm not familiar with the name. Is it more spinach like, kale/cabbage/mustard like?

Thank you.

I'm not sure about the green element. But enoki mushrooms are very delicate..and probably should be cook separately on low heat with just a light tossing about the pan with some oil, ghee, or light veggie oil/salt...aji mirin.
They'll probably break apart into nothing by the time the greens are cooked if done together.

Taste the green, if it's like a mustard green/kale/collard green. Some bacon fat..or sautee with some fat rendered from pork belly and then finish the pork belly in the oven and plate with the greens, topped with the mushrooms. Some prosciutto ham, or other Italian ham could work also.