Heat oil in a wok over medium-high heat. Cook and stir onion, mushrooms, and garlic in hot oil until the mushrooms are soft, about 5 minutes. Add butter and vinegar to the mushroom mixture; season with salt and pepper. Toss mushroom mixture in the wok until the butter melts and coats everything, about 1 minute.

Switch heat off and add kale to the wok; cook and stir in the hot wok until the kale is bright green, about 5 minutes. Season kale mixture with nutmeg and stir.