Combine brown sugar, chili powder, paprika, garlic powder, kosher salt, black pepper, and cayenne pepper in a small bowl. Rub the dry rub mixture all over the trimmed shoulder roast. Pierce roast with a fork to help seasoning penetrate meat. Wrap with plastic wrap and refrigerate, 8 hours to overnight.

Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Add olive oil. Cut pork into 4 to 6 pieces and add to the hot oil. Cook a few pieces at a time until browned, about 5 minutes. Transfer cooked meat to a plate.

Add chicken broth to the pot and deglaze the browned bits from the bottom by stirring constantly with a whisk until smooth. Turn Instant Pot® off.

Add water, apple cider vinegar, 2 tablespoons brown sugar, 2 tablespoons tomato paste, onion, and 2 tablespoons ketchup. Top with browned pork. Close and lock the lid, making sure the vent is closed. Select high pressure according to manufacturer's instructions; set timer for 60 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Remove meat and onions and cover to keep warm.

Set Instant Pot® to saute and add remaining tomato paste, remaining ketchup, and 1/2 cup brown sugar. Use a ladle to skim excess fat from the top of the liquid and discard it. Cook until sauce is reduced in volume by about 1/2, about 15 minutes.

Meanwhile, use 2 forks to shred cooked pork into small chunks. Serve meat as desired, topped with the thickened sauce.