Heat a drizzle of neutral oil in a skillet over medium-high heat. Add the bulgogi and cook until browned and fully cooked through, about 3–4 minutes. Remove and set aside.

In the same pan over medium heat, add the chopped kimchi. Cook, stirring occasionally, for 7–10 minutes until translucent and lightly browned at the edges. If the pan gets too dry, add reserved kimchi juice one tablespoon at a time.

Stir in the sugar and sesame oil toward the end. Remove from heat and add to the bowl with the bulgogi.

Use kitchen scissors to snip the bulgogi and kimchi together directly in the bowl into bite-sized pieces. Alternatively, transfer to a chopping board and chop. Set aside.

Heat a clean pan over medium-low heat. Add ½ tablespoon of butter and lay one slice of sourdough over it. Move the bread around the pan to absorb the butter evenly.

Layer 2 slices of American cheese on the bread, then add all the kimchi and bulgogi filling, then top with the remaining 2 slices of cheese.

Place the second bread slice on top. Spread the remaining ½ tablespoon of butter directly on the top of the bread. Cook for 3–4 minutes until the bottom is deep golden brown, then carefully flip. Cook another 3–4 minutes until golden and all the cheese is fully melted.

Transfer to a cutting board, slice diagonally through the center, and serve immediately.