Spread the gochujang on one side of each of 2 slices of bread.
Add 2 tablespoons butter to a large nonstick pan over medium-high heat. (If the butter begins to brown immediately, turn down the heat a bit.) Swirl the melting butter around the pan to coat. Add the 2 pieces of prepared bread to the pan, gochujang-side up. Move them around to absorb the melted butter. Layer 2 pieces of cheese on each slice and top each with the final 2 pieces of bread. Cook on one side until a peek at the underside reveals a golden brown crust, then carefully flip, adding the remaining 2 tablespoons butter to the pan and tucking it underneath the sandwiches. Cook until the other side is golden.