Heat sesame oil in a small skillet over medium heat. Add squeezed chopped kimchi and cook for 3 to 4 minutes, stirring, until no free liquid remains. Transfer to a plate for 2 minutes.
Divide half the Gouda over 2 bread slices, spread cooked kimchi evenly over it and add remaining cheese. Close with the other bread slices.
Spread mayonnaise thinly over the outside faces. Heat a large skillet over medium-low heat.
Cook sandwiches for 4 to 5 minutes per side until golden and the Gouda is fully melted, covering for 2 minutes after turning if the center needs help warming.
Rest for 1 minute, cut in half and serve.