For the pork: Preheat the oven to 300 degrees F. Place the pork shoulder into a small baking dish. Coat all over with the canola oil and sprinkle liberally with salt and pepper. Cover tightly with aluminum foil and roast until the pork is very tender and an instant-read thermometer inserted into the thickest part registers 190 degrees F, about 3 hours.
Remove the pork from the oven and let it rest for 30 minutes. Reserve 1/4 cup of the roasting juices, then discard all of the fat and remaining juices from the baking dish. Use 2 forks to tear the meat into bite-size pieces. Whisk the barbeque sauce and apple cider vinegar together in a small bowl. Add 1 cup of the barbeque sauce to the baking dish and toss to combine with the pork. Reserve the remaining sauce for drizzling on the pizzas. Cover the pan loosely with the foil and set aside. Increase the oven to 500 degrees F and place a rack on the bottom shelf.
For the marinara: Heat the olive oil in a medium saucepan over medium heat. Add the onions and cook until softened, stirring occasionally, 8 to 10 minutes. Add the garlic and cook until softened, stirring occasionally, 3 to 5 minutes. Add the sugar, oregano, crushed tomatoes and some red pepper flakes. Stir to combine and season liberally with salt and pepper. Bring to a boil over medium-high heat then reduce to a simmer. Simmer, uncovered, until the sauce has thickened and is reduced to 2 cups, about 30 minutes. Discard the oregano and season with salt and pepper, if needed. Cover with a lid and reserve.
For the pizzas: Cut a piece of parchment paper to fit on the back of an inverted baking sheet. Place the inverted baking sheet in the oven on the bottom shelf. Lightly dust the parchment with flour. Stretch or roll 12 ounces of the pizza dough into an 11-inch round, about 1/8-inch thick, directly on the parchment paper. Spread 1/2 cup of the marinara sauce evenly on the pizza, leaving a 3/4-inch border.
Sprinkle 1 1/2 cups of the mozzarella all over the marinara sauce. Spread 1 cup of the pulled pork on the cheese, making sure that each bite gets a piece of meat. Flatten the meat with your fingers as you place it on the pizza. Drizzle the entire pizza with 1/4 cup barbeque sauce. Brush the crust with the olive oil.
Carefully lift the parchment and pizza onto the inverted baking sheet in the oven. Cook until the crust is golden brown and the cheese is bubbly, 12 to 14 minutes. Remove from the oven. Allow to cool on the inverted baking sheet for 5 minutes before serving. Repeat the process with the remaining ingredients to make 3 additional pizzas.