In a large nonstick skillet, melt 2 tablespoons of the butter. Add the mushrooms, season with salt and pepper and cook over moderate heat, stirring occasionally, until softened and lightly browned, about 5 minutes.
In a bowl, season the eggs with salt and pepper. Melt the remaining butter in the skillet with the mushrooms; add the eggs. Cook over moderately low heat without stirring until the bottom is barely set, 30 seconds. Add the chives and cook, stirring occasionally, until the eggs form large soft curds. Remove from the heat and sprinkle on the cheese; let stand until softened, 30 seconds. Gently fold the cheese into the eggs; serve with toast.