Preheat the oven to 300 F.
In a small bowl, combine the dry rub ingredients. Pat and rub all sides of the pork shoulder with the dry rub.
Heat olive oil in a Dutch oven over medium heat. Once hot, sear all side of the pork shoulder, about 1-2 minutes per side.
Add chicken broth to the pot, cover, and place in the oven. Cook for 2 to 3 hours. Check the pot at the 90-minute mark to see if you need to add more broth. Use a meat thermometer to check the internal temperature of the roast. When the pork reaches an internal temperature of 190 F to 195 F, the meat is ready to be shredded.
While the pork is cooking, place the chipotle aioli ingredients into a food processor and processes for 1 minute. Place into the refrigerator until ready to use.
Remove the pork from the pot and set aside on a large cutting board. Using a large spoon, remove and discard most of the fatty liquid floating on top. Leave the broth in the pot.
Add the tomato paste, brown sugar, Worcestershire, apple cider vinegar, and smoked paprika to the pot with the broth. Whisk all ingredients together. Heat on medium-low and simmer for 10-15 minutes, until BBQ sauce thickens up slightly.
Shred the pork with two forks. Place the shredded pork into the pot with the homemade BBQ sauce. Toss to coat an set aside.
Increase the oven temperature to 425 F.
Lay out tater tots in a single layer on a baking sheet. Place into the oven and bake for 20 minutes, flipping them once, half way through cooking. Add shredded cheese 2 minutes before baking time is done.
Remove tater tots from the oven. Evenly spread out the BBQ pulled pork on top of the tots. Drizzle with chipotle aioli and garnish with green onions. Serve immediately.