Cut onions into ½-inch slices. Heat a large sauce pan over medium-high. Add oil to pan, cook onions for 2 minutes, do not brown. Stir in salt, turn heat down to low. Cook onions until they’re caramelized, stirring occasionally, about 30 minutes.
Generously season steak of your choice with salt and pepper. Grill until rare, rest, and slice ½-inch thick across the grain. Lightly grill cut side of bread, spread with mustard, layer with steak, onions, cheese, a drizzle of vinegar, and arugula. Cut sandwich into portions, serve with home fries, more vinegar for dipping.