Stack the cheeses starting with a slice of cheddar, then a slice of American, then a slice of cheddar and set the cheese stack aside to allow it to come to room temperature.
In a pan that is large enough to toast the sandwich later, cook 1 bacon weave or 3 slices of bacon.
When the bacon is nearly crisp, set aside on a paper towel-lined plate and leave the bacon fat in the pan for later use.
Drain and coarsely chop 4 tablespoons of kimchi.
Spread mayonnaise on both sides of 2 slices of white bread.
Preparation
Reheat the skillet over medium heat.
Brown one side of each slice of bread and place the bread, browned side up, on a work surface.
Quickly, place the cheeses on one of the slices of bread, and place the second slice of bread over the cheese, browned side down.
Saute the kimchi in the skillet and until it starts to caramelize, about 3-5 minutes.
Remove the top piece of bread from of the cheese and cover the cheese with the hot kimchi.
Place the bacon on top of the kimchi, and place the bread browned side down, over the bacon.
Reduce the heat to medium-low and use a spatula to carefully place the sandwich into the pan.
Cover the sandwich with a press, or press down on the sandwich with a spatula as it cooks.
When the first side of the sandwich is golden brown, carefully flip it over and cover with the press until the bottom side is golden brown.
Slice the sandwich in half, diagonally, and serve immediately.