Combine the onion soup mix, au jus mix, and Italian dressing packets in a small bowl. Whisk the mixture together to ensure even distribution.

Pat the chuck roast completely dry with paper towels, then rub the seasoning mixture all over the meat, pressing gently to help it adhere. Place the seasoned roast in your slow cooker.

Pour the beef broth and water around the roast, not over it. The liquid should come about halfway up the sides of the meat.

Cover and cook on low for 6-8 hours, until the meat easily shreds with a fork.

Remove the roast and shred it using two forks, discarding any excess fat. Strain the cooking liquid through a fine-mesh strainer into a bowl, pressing the solids to extract maximum flavor.

Lightly butter and toast the hoagie rolls until golden brown. Layer the shredded beef generously on the bottom halves, top with cheese, and place under the broiler for 2-3 minutes until the cheese bubbles. Serve immediately with warm au jus in small bowls for dipping.