Drain the kimchi thoroughly. Squeeze it gently with clean hands or press it in a sieve to remove excess liquid. Roughly chop it into shorter bite-size pieces. If the kimchi is very juicy, place it in a dry skillet over medium heat for 1–2 minutes, just long enough to steam off excess moisture. Do not cook it until soft or browned — the goal is only to dry it slightly. If needed, add the 1 tsp sugar and stir briefly to round out the acidity. Transfer to a plate and let it cool slightly. The kimchi should be flavorful and still vivid red, but not dripping with juice. This is what keeps the bread from steaming and going soggy. Kimchi's fermentation process produces genuinely more liquid, and more aggressively flavoured liquid, than a typical raw vegetable would release — squeezing and pressing addresses the bulk of that moisture mechanically, while the brief dry-skillet step handles whatever remains without pushing the kimchi toward actually cooking down.

Lay out the bread slices. Butter one side of every slice evenly, all the way to the edges. Turn 4 slices over so the unbuttered side faces up. Divide about half of the mozzarella between those 4 slices. Spread it in an even layer. Top with the drained chopped kimchi, keeping it mostly in the center and not overloading it. Season very lightly with black pepper. Top with the remaining mozzarella, then close each sandwich with the remaining bread slices, keeping the buttered side facing outward. Cheese should sit both below and above the kimchi. This traps the kimchi in place and helps the sandwich hold together when the cheese melts. Sandwiching the kimchi between two cheese layers rather than placing it directly against either slice of bread does genuine double duty — it anchors the filling structurally once the cheese melts, and it acts as a moisture barrier protecting the bread from whatever liquid the kimchi still carries even after draining.

Heat a skillet or griddle over low to medium-low heat. Place the sandwiches in the pan and cook in batches if needed. Cook for 3–5 minutes per side, pressing very lightly with a spatula only to encourage even contact. Adjust the heat as needed so the bread turns deep golden and crisp without burning before the cheese melts. If the bread is coloring too quickly before the cheese melts fully, lower the heat and cover the pan loosely for 30–60 seconds to help the heat penetrate. Flip carefully and cook the second side until equally deep golden and the mozzarella is fully melted and stretchy. The bread should be deeply golden, evenly crisp, and buttery, while the cheese is completely melted. The sandwich should feel crisp outside and soft, gooey, and cohesive inside. This low, patient heat is calibrated specifically around the substantial amount of cheese and kimchi filling packed into the sandwich — a hotter pan would brown the bread well before that filling has had time to fully melt through and heat.

Transfer the sandwiches to a rack or board and let them rest for 1 minute so the cheese settles slightly. Slice each sandwich in half. For the best presentation, stack the halves so the kimchi and mozzarella cross-section shows clearly. Resting briefly prevents the filling from sliding out too aggressively while still preserving the dramatic cheese pull. Cheese straight off the griddle is at its most fluid; this brief pause lets it firm up just enough to hold together cleanly when cut, without cooling so much that it loses its stretch.