Slice the onion into thin half-moons.
Heat half of the olive oil in a skillet over medium-low heat.
Add the onions and cook slowly, stirring occasionally, until they are golden and soft.
Stir in the balsamic vinegar during the last minute of cooking the onions, then remove from heat.
Season both sides of the flank steak with sea salt and black pepper.
Heat the remaining olive oil in a cast-iron skillet over medium-high heat.
Sear the steak for 3 to 4 minutes per side for medium-rare.
Transfer the steak to a cutting board and let it rest for 5 minutes.
Toast the sourdough bread until it is golden brown.
Slice the steak thinly against the grain.
Spread the Dijon mustard on the toast, then layer with arugula, steak slices, and caramelized onions.