Season the shaved beef with salt and pepper.

Heat olive oil in a pot over medium heat. Sauté onions until softened and starting to brown. Add beef broth, beef, and rosemary, and bring the mixture to a simmer. Let it cook for 15-20 minutes, allowing the flavors to concentrate.

Stir in the butter and taste. Add more salt and pepper to your liking. Be sure the beef reaches an internal temperature of 145°F (63°C) for medium-rare or 160°F (71°C) for medium.

Preheat the oven broiler to low. Slice open the rolls and place them on a baking sheet. Toast the rolls under the broiler until golden brown, keeping a close eye to prevent burning.

Using tongs or a slotted spoon, place the beef on the toasted rolls. Serve with the reserved au jus on the side for dipping.

Mix together the lemon juice, water, and sugar for the lemonade in a pitcher. Stir well until the sugar dissolves. Serve over ice if desired.

Serve the French dip sandwiches with potato chips, pickle spears, and the homemade lemonade.