Add two Tablespoons of butter to a medium-sized sauté pan, and increase the heat to medium-high.
When the butter is melted, add the sliced onion, white granulated sugar, and Kosher salt.
Allow the onions to cook, while stirring every few minutes to ensure that they cook evenly.
*If the onions are browning too quickly, add a Tablespoon or two of water to the pan, and stir to combine.
When the onions are golden-brown, remove from the heat and allow them to cool down.
Use the remaining butter to spread on both sides of each piece of sourdough sandwich bread.
Griddle the first side of each slice of bread until golden brown.
*Note that this first toasted side will go on the inside of your sandwich.
With the griddled sides of the sandwich bread facing inwards, assemble the sandwich with the Monterey Jack cheese, Gruyère cheese, sliced apple, and the caramelized onions from earlier.
Griddle the outsides of the sandwich on both sides over medium-low to medium heat, adding a lid during the last minute or two of cooking to ensure that the cheese is fully melted on the inside of the sandwich.
*Avoid cooking the sandwich over high heat, as the outsides will darken too quickly and the cheese won’t be fully melted.
Cut the sandwich in half on a diagonal, creating two triangles.
Serve immediately, with extra napkins.