Rub the pork shoulder with the chile powder. Place in a large, lined slow cooker. Pour the cola around the pork. Cover and cook on low heat until the meat is fork-tender and cooked through, about 8 hours.
Using two large forks, transfer the pork to a cutting board and let cool slightly. Shred the meat.
Pour the cooking liquid into a large bowl or 1-qt. measuring cup and let stand for 10 minutes. Skim the fat from the surface and discard.
Return the meat to the cooker and warm over low heat, mixing in some of the cooking liquid to moisten.
To serve, pile the pulled pork onto the bun bottoms and top with pickles. Drizzle with your favorite barbecue sauce and cover with the bun tops.