Heat the oil in a pot on medium heat. Saute until the onions soften and start to brown. Then stir in the broth, beef, and rosemary. Bring to a low simmer until the flavor concentrates, about 15 to 20 minutes. Then add in the butter. Taste and season with salt and pepper to taste.
Prep the carrots.
Preheat the oven broiler on low. Slice open the rolls and place them on a baking sheet. Toast the bread under the broiler until golden brown (keep a close eye on the bread so that it doesn't burn).
Using tongs or a slotted spoon, place the beef on the roll. Optional: place the cheese over the beef and melt under the broiler.
Serve the sandwich with a side of au jus for dipping. Enjoy carrot sticks with ranch and lemonade.