Season the shaved steak with salt and pepper.

Heat the olive oil in a pot over medium heat. Sauté the onions until softened and starting to brown.

Stir in the beef broth, seasoned steak, and rosemary. Bring to a low simmer and cook for 15–20 minutes until the flavor concentrates. Stir in the butter, then taste and adjust the seasoning. Reserve the au jus for dipping.

Preheat the oven broiler on low. Slice open the rolls and place them on a baking sheet. Toast under the broiler until golden brown, watching carefully to avoid burning.

Using tongs or a slotted spoon, place the beef onto the toasted rolls. Optional top with cheese and melt under the broiler.

Mix the lemon juice, water, and sugar to taste in a pitcher. Add ice cubes if desired.

Serve the French dip sandwiches with au jus for dipping, alongside potato chips, pickle spears, and homemade lemonade.