Melt 1 tablespoon of the unsalted butter in a 10-inch cast iron or oven- and broiler-safe nonstick skillet over medium-high heat. Add the mushrooms and remaining 1/2 teaspoon kosher salt, spread into an even layer, and cook undisturbed for 2 minutes. Toss and continue to cook until golden-brown and some of their liquid is released, tossing every 2 minutes, for 4 to 6 minutes more. Add the scallion whites, thyme, and remaining 1 tablespoon unsalted butter. Cook, stirring occasionally, until the scallions soften and begin to brown and the mixture is fragrant, about 2 minutes.