Pulled Pork Quesadilla with Chipotle-Cola BBQ Sauce and Avocado Cream

12-14 (8-inch) flour tortillas

7 cups shredded sharp cheddar cheese

For the Rub:

2 tablespoons brown sugar

1 tablespoon kosher salt

1 tablespoon smoked paprika

1 teaspoon ground cumin

1 teaspoon chili powder

5 1/2 to 6 pound bone-in pork shoulder roast (Boston Butt), lightly trimmed

3 large mesquite wood chunks, soaked in water for at least 30 minutes, or, for gas grills, 3 handfuls wood chips

For the Sauce:

2 tablespoons unsalted butter

1 medium onion, finely chopped

1 1/2 cups room temperature cola

1 1/3 cups ketchup

1/3 cup cider vinegar

3 tablespoons brown sugar

1 tablespoon chili powder

2 teaspoons smoked paprika

1 1/2 teaspoons garlic powder

1/2 teaspoon ground chipotle pepper

Avocado Cream:

1 ripe Hass avocado

1/2 cup sour cream

1/2 teaspoon kosher salt

1 teaspoon fresh lime juice

1. Make the rub: combine the sugar, salt, smoked paprika, cumin, and chili powder in a small bowl; rub evenly over the pork. Let stand at room temperature 45 minutes.
2. Set up grill for medium-low indirect heat cooking (275°- 300°F). Place an aluminum pan half filled with water, beneath the food rack directly under where the pork will be.
3. Place a wood chunk on the lit charcoal (For a gas grill, place wood chips in a smoke box). Set the pork, fat side up, on the grill; close the lid. Grill to an internal temperature of 160°F, about 3 1/2 to 4 hours, adding a wood chunk and charcoal or wood chips as needed (about every hour) to maintain the grill temperature.
4. Wrap the pork in heavy-duty aluminum foil and return to the grill over indirect heat; close the grill. Cook about another 1 to 2 hours, until the pork reaches 195°F. Remove from the grill and let rest in foil for 30 minutes.
5. Meanwhile, melt the butter in a medium saucepan over medium heat and add the onion; cook, stirring occasionally, until softened, about 6 minutes. Stir in the cola, ketchup, vinegar, sugar, chili powder, smoked paprika, garlic powder, and ground chipotle and bring to a boil over medium-high heat. Reduce the heat to medium and simmer until slightly thickened, 15-18 minutes.
6. Make the avocado cream: puree the avocado, sour cream, salt, and lime juice in a food processor.
7. Unwrap the roast and, when cool enough to handle, shred the meat, discarding any large pieces of fat. Combine with sauce.
8. Working one at a time, place a tortilla on a work surface and top half with 1/4 cup cheese. Spoon 3/4 cup pork over the cheese; top with another 1/4 cup cheese. Fold the tortilla over the filling to form a semi-circle. Repeat with remaining ingredients.
9. Heat a large nonstick skillet over medium-low. Add 2 quesadillas and cook, turning once, until the tortilla is lightly browned and the cheese is melted, about 2-3 minutes per side. Repeat with remaining quesadillas. To serve, cut quesadillas in half and top with the avocado cream.