Pat the pork chops dry and season all over with 1 teaspoon of the salt. Heat the oil in a large skillet over high heat. Once the oil is glistening, add the pork and cook undisturbed until browned, about 3 minutes. Flip and continue cooking until the internal temperature of the pork reaches 145°F on an instant-read thermometer, 3 to 4 minutes more. Transfer the pork to a plate and let the skillet cool slightly.
Heat the skillet over medium heat and add the mushrooms in a single layer. Season with the remaining ½ teaspoon salt and cook undisturbed until they begin to brown, about 4 minutes.
Stir in the butter, miso paste, garlic, and ginger until coated. Cook, stirring often, until mushrooms are tender, about 3 minutes more. Stir in the heavy cream, cilantro and green onions until combined.