Preheat your oven to 300°F (150°C) if roasting, or prepare your slow cooker according to manufacturer instructions. Mix paprika, garlic powder, salt, and black pepper to create a simple spice rub.
Rub the spice mixture all over the pork shoulder, pressing gently to help it adhere. Let it sit for a few minutes while you prepare the cooking setup.
Place the pork shoulder on a baking sheet or into your slow cooker. Drizzle with vegetable oil or pork drippings to help the rub stick and to add moisture during cooking.
If roasting, cover the pork tightly with foil and roast for about 3-4 hours until the meat is fall-apart tender, or cook in the slow cooker on low for 6-8 hours. The meat should reach an internal temperature of 195°F-205°F and be easily shredded.
Once cooked, transfer the pork to a large platter or cutting board. Carefully shred the meat with two forks, tearing it into sticky, glistening strands that ooze with juices and smoky glaze. It should be tender and easy to pull apart.
Mix the shredded pork with barbecue sauce, working it into every strand to ensure every bite is flavorful and sticky. Adjust the amount of sauce to taste, adding gradually for balance.
Toast the brioche buns lightly if desired, or serve them soft and warm. Prepare sliced red onions and pickles for garnishing.
Assemble each slider by piling a generous amount of pulled pork onto the bottom half of a bun. Top with thinly sliced onions and pickles, then cover with the top bun.
Serve immediately, allowing the juices to soak into the bun and the flavors to meld. Enjoy the messy, smoky satisfaction of these delicious pulled pork sliders!