Rub rump roast all over with salt and pepper. Let sit at room temperature for at least 30 minutes or up to 1 hour.

Meanwhile, whisk together beef broth, water, onion soup mix, and Worcestershire in a 7-qt. slow cooker.

Heat oil in a large cast-iron skillet over medium-high. Add roast and cook until well browned on both sides, 3 to 4 minutes per side; transfer to slow cooker.

Reduce heat to medium and add onions to skillet; cook, stirring occasionally, until softened and beginning to caramelize, 8 to 10 minutes. Transfer onions and any accumulated juices to slow cooker.

Cover and cook until roast is tender and shreds easily, about 5 hours on HIGH or 8 hours on LOW.

Preheat oven to 375°F. Transfer roast to a large bowl and shred with two forks.

Pour broth mixture through a fine mesh strainer into a large heatproof bowl; transfer onions and 1 cup of the strained broth to bowl with shredded roast and stir to combine. Reserve remaining strained broth (au jus) for later use.

Using a serrated bread knife, cut rolls in half horizontally. Place rolls, cut sides down, on a large rimmed baking sheet. Bake in preheated oven until lightly toasted, about 8 minutes.

Remove rolls from oven; flip cut side up. Arrange 1 cup roast mixture over each bottom half of rolls; top each with 2 overlapping slices of cheese.