Prep all ingredients per instructions above.
Preheat the oven to 325°F (163°C) or use a slow cooker.
Mix together the brown sugar, paprika, salt, black pepper, chili powder, garlic powder, onion powder, and cayenne pepper to make the dry rub.
Pat the brisket dry with paper towels. Rub the dry rub all over the brisket, covering all sides. Wrap tight in foil (if using a slow cooker skip the foil and place brisket into the slow cooker).
Place the brisket on a baking sheet fat side up. Cook for 3 hours.
If using a slow cooker add broth about ¾ of the way up the brisket and cook 6 - 8 hours on low. Then set aside to rest for about 30 minutes before slicing.
Place the potatoes in a pot of water. Bring the water to a boil, then lower the heat and simmer for 10 to 15 minutes, until the potatoes are fork-tender. Drain the water completely from the pot. Set aside to cool.
In a bowl mix the mayo, mustard, pickle, dill, celery, onion, and carrots. Mix in the potatoes and season with salt and pepper to taste. Add more mayo to the preferred level of creaminess. Refrigerate until ready to serve.
Once ready to serve, warm the barbecue sauce in a saucepan over low heat.
Slice the brisket against the grain into thin slices.
Assemble the Texas brisket sandwich by placing slices of brisket on the hamburger buns, topping with barbecue sauce, and optional pickle slices. Serve with potato salad and a cold amber ale.