For the burrito: In a large pan on medium-high heat, add the adobo sauce, barbecue seasoning, chicken, 2 tablespoons of the oil and the scallion whites. Saute until everything is combined and coated and the chicken is warmed through, 6 to 8 minutes. Remove to a large bowl.
In a medium bowl, toss the rice with the lime zest and scallion greens; season with a pinch of salt and a few grinds of black pepper.
Drain the beans over a bowl and reserve 1 cup of the liquid for the topping. Add the beans to the bowl with the chicken mixture, then add the rice mixture and cheese and mix to combine.
Add a couple of spoonfuls of filling to each tortilla. Fold one of the tortillas by first overlapping the line of filling with the flap closest to you. Tuck in both sides and, while holding them, fold the tortilla over itself, keeping the ingredients tucked tightly in the center. Fold the remaining tortillas.
Preheat the oven to broil.
On a griddle or in a large pan on high heat, add the remaining tablespoon of oil. Add each burrito seam-side down and sear until golden brown, 2 to 3 minutes (you may have to do this in batches if using a pan instead of a griddle). Flip and sear the other side, another 2 to 3 minutes. Transfer the burritos to an ovensafe serving dish.
For the topping: Drizzle the reserved 1 cup bean liquid over the burritos, top evenly with the cheese and place under the broiler until the cheese is just melted, 3 to 5 minutes. Serve topped with the white onions, cilantro and a squeeze of lime.