Preheat the oven to 225°F. Set the pork on a cutting board and score the fat in a crosshatch pattern, making the cuts about ¾” deep and 1” apart.
To make the rub, combine the pimento, garlic powder, cayenne pepper, brown sugar, celery salt, mustard powder and 1½ tsp salt in a small bowl. Massage it into the meat.
Place the pork in a roasting pan and pour the beer into the pan. Cover with foil and roast for 3 hrs. Check it periodically and turn the roast over every hour.
Remove foil and continue roasting until the pork is 190°F, about 1½ more hours. Remove from the oven, cover loosely with foil and let rest 30 mins.
To make the sauce, combine the vinegar, garlic, chile, brown sugar, tomato paste, Dijon mustard and 1½ tsp salt in a small saucepan and bring to a boil; set aside and remove the chile.
Pull the pork into long shreds using 2 forks and rid of the fat as desired. Toss the pork with any residual pan juices and add salt to taste.
Serve the pork on slider rolls with a drizzle of sauce; top with coleslaw and sliced cucumbers.