Melt one more piece of butter in the skillet and reduce heat to medium-low. Add sandwich and cook, swirling occasionally, until deep, even golden brown, about 5 minutes. Remove the sandwich using a flexible metal spatula.
Add the remaining butter. Return sandwich to skillet cooked-side up—season with salt. Cook, occasionally swirling, until the second side is deep, even golden brown, and the cheese is thoroughly melted for about 5 minutes. Serve immediately.