Thinly slice the yellow onion into uniform half-moons.
Heat the olive oil in a non-stick skillet over medium-low heat and add the onions with a pinch of the sea salt.
Sauté the onions slowly for 15-20 minutes, stirring occasionally, until they reach a deep golden brown color.
Stir in the balsamic vinegar to deglaze the pan, then remove the onions and set them aside.
Wipe the skillet clean and return it to the stove over medium-high heat.
Form the ground beef into a single large, flat patty and season both sides with the remaining sea salt and black pepper.
Sear the beef patty for 3-4 minutes per side until a dark crust forms and it is cooked to your desired level of doneness.
While the beef cooks, toast the sourdough bread slices until they are golden and firm.
Spread the Dijon mustard evenly across one slice of toasted bread.
Assemble the sandwich by layering the fresh arugula, the beef patty, and the caramelized onions between the sourdough slices.