Prepare the caramelized onions: Melt the butter in a skillet over medium-low heat. Add the sliced onions and a pinch of salt. Cook slowly for 15-20 minutes, stirring occasionally, until deeply browned and soft. Stir in brown sugar (if using) during the last 5 minutes.

While the onions are cooking, prepare the cheese blend by mixing the shredded sharp cheddar and Gruyère in a small bowl.

Assemble the sandwiches: Spread a thin layer of Dijon mustard on the inside of all four slices of sourdough bread. Top two slices with half of the cheese mixture, followed by an even layer of caramelized onions, then the remaining cheese. Top with the remaining bread slices.

Prepare the exterior: Spread a thin, even layer of mayonnaise on the outside of all four slices of bread instead of butter. This creates a perfectly golden crust.

Grill the sandwiches: Heat a large non-stick skillet or griddle over medium heat. Place the sandwiches on the hot surface and cook for 3-4 minutes per side, pressing gently with a spatula, until the bread is golden brown and the cheese is completely melted and gooey.

Remove from heat, slice diagonally, and serve immediately, perhaps alongside a bowl of warm tomato soup.