For the brisket: Combine the salt, pepper, paprika, cayenne, cumin, granulated garlic and dry mustard in a small bowl. Sprinkle the brisket on all sides with the spice rub. Place the brisket into a slow cooker and cover with the lid. Set the slow cooker to high and cook until tender, about 6 hours.
Remove the brisket from the slow cooker and set aside until cool enough to handle. Pour the liquid from the slow cooker into a fat separator. Discard the fat.
For the BBQ sauce: Pour the cooking liquid into a stockpot and place over medium heat. Stir in the tomato paste, brown sugar, vinegar, salt, liquid smoke, cayenne, granulated garlic and dry mustard. Bring to a simmer, then reduce the heat to medium-low. Simmer, stirring occasionally, until thickened, at least 20 minutes.
Slice the brisket into 1-inch chunks. Remove and discard any large pieces of fat. Spread the brisket in a 9-by-13-inch pan and pour the sauce over the meat.
Turn the broiler on high. Broil until crispy and slightly charred on the edges, about 5 minutes. Serve warm with white bread or rolls.