In a small bowl, combine the dry rub ingredients. Pat and rub all sides of the pork shoulder with the dry rub.

Heat 2 tbsp of olive oil in a skillet. Once hot, sear all side of the pork shoulder just until golden and crisp, about 2 minutes per side. Transfer to a slow cooker and cook on high for 4 hours or on low for 8 hours until cooked all the way through.

Remove from slow cooker and set aside to rest before shredding. Using a large spoon, remove and discard the fat liquids floating on top from the liquids remaining in the slow cooker.

Place remaining non-fatty liquids into a medium saucepan over medium heat. Add tomato paste, apple cider vinegar, smoked paprika, brown sugar, and Worcestershire. Whisk all ingredients together. Once it comes to a boil, reduce heat to medium-low and simmer for 15 minutes. Sauce will thicken up a bit.

Using two forks or your hands, shred the cooked pork. Toss the shredded pork with the homemade BBQ sauce and set aside.

Preheat oven to 425 degrees F.

Lay out tater tots in a single layer on a baking sheet. Drizzle remaining 2 tbsp of olive oil to coat the tater tots. Place into the oven and bake for 20 minutes, flipping them once, half way through baking. Add shredded cheese 2 minutes before baking time is done.

While the tater tots are baking, place chipotle aioli ingredients into a food processor and processes for 1 minute.

Remove tater tots from the oven. Evenly spread out shredded BBQ pulled pork on top. Drizzle with chipotle aioli and garnish with green onions. Serve immediately.