Preheat the oven to 300℉ (148℃). (Alternatively, use a slow cooker and cover the brisket ¾ of the way with broth or water.)
In a bowl, mix together brown sugar, paprika, salt, black pepper, chili powder, garlic powder, onion powder, and cayenne pepper to make the dry rub.
Pat the beef brisket dry with paper towels. Rub the dry rub all over the brisket, covering all sides. Wrap in foil and place fat side up on a baking sheet. Cook the brisket in the oven for 3 hours (or in the slow cooker for 6–8 hours). Turn off the oven and let the brisket rest for about 30 minutes before slicing.
Preheat the oven to 400°F (204°C). Line a baking sheet with parchment paper or lightly grease it.
In a large bowl, combine olive oil, garlic powder, paprika, dried oregano, dried thyme, salt, and black pepper. Toss the potato wedges in the seasoning mixture until evenly coated.
Arrange the wedges in a single layer on the baking sheet, ensuring they arent overlapping. Bake for 25–30 minutes, flipping halfway, until golden brown and crispy on the outside and tender inside.
Steam the broccoli: Bring a pot of salted water to a boil. Add broccoli florets, reduce heat, and simmer for 5–7 minutes until bright green and tender. Drain and season with salt and pepper. Set aside.
Slice the brisket thinly against the grain.
Assemble the sandwiches: Place sliced brisket on the bottom bun and spoon barbecue sauce on top (if using). Cover with the top bun.
Serve the Texas brisket sandwiches with potato wedges, steamed broccoli, ketchup, and cold IPA beer.
Save or freeze any leftover brisket for future meals.