Heat an extra-large, heavy skillet over medium heat until hot, about 3 minutes. Meanwhile, pat the brisket dry, cut into 6 equal pieces and place in a large bowl. In a small bowl, mix together the chile powder, 1 tsp. salt and 1/2 tsp. pepper. Drizzle the EVOO over the meat and toss to coat. Sprinkle with the chile powder mixture and toss again. Add half of the meat to the hot skillet and cook, flipping once, until browned, 7 to 8 minutes. Transfer to a large, lined slow cooker. Repeat with the remaining meat.
Pour 1 cup cola into the skillet to deglaze the pan, scraping up any browned bits with a wooden spatula. Pour the cola mixture and tomatoes into the slow cooker. Cover and cook on high until the meat is tender, about 6 hours. Wash the skillet.
Transfer the meat to a plate and let cool. Pour the juices from the cooker into a 1-qt. glass measuring cup and let stand until the fat rises to the top. Skim the fat from the juices and discard. Pour the juices into the cleaned skillet. Bring the juices to a boil over high heat, cooking until reduced by half, about 15 minutes. Shred the meat with 2 forks. Add to the skillet; toss with the juices and remaining 1/2 cup cola. Season to taste with salt and pepper. Cover to keep warm.
Preheat the broiler. Open the rolls but do not separate, then place cut side up on 2 baking sheets. Spread with the mayo. Place 2 pickle slices on each roll bottom. Lay a slice of cheese, folded lengthwise, across the entire roll. Broil the rolls, 1 sheet at a time, until the cheese melts, about 2 minutes. Repeat with the remaining sheet of rolls. Using tongs, transfer the meat to the buns. Fold up and serve.