Pat the chuck roast dry. Rub with the mustard, then season all over with the salt, pepper, and 2 tablespoons of the barbecue seasoning.

Smoke until the roast develops a deep, dark bark and reaches an internal temperature of 180°F/80°C, 2 to 3 hours.

Transfer the roast to a disposable aluminum pan or baking dish. Pour in the beef broth, sprinkle with the remaining 1 tablespoon barbecue seasoning, cover tightly with foil, and return to the smoker. Cook until the beef is very tender and shreds easily with a fork, 2 to 3 hours more.

Meanwhile, combine the pepper Jack, Cheddar, cream cheese, tomatoes with their juices, jalapeños, onion, and milk in a cast-iron skillet or disposable aluminum pan.

Place the queso on the smoker and cook, stirring occasionally, until the cheeses are melted and the dip is smooth, 30 to 45 minutes.

Finely chop or shred the beef and stir it into the queso until evenly combined. If the dip is too thick, stir in a little more milk until it reaches the desired consistency.

Serve hot with tortilla chips or toasted bread.