In a large bowl, toss the shredded chicken with 1 cup barbecue sauce and 1 cup of the cheese.
Bake:
Place the tostadas on a rimmed baking sheet. Distribute the chicken mixture evenly over the tostadas. Bake for 6 to 8 minutes, until the cheese has melted.
Serve:
Top the tostadas with the remaining 1 cup cheese, drizzle with remaining ½ cup barbecue sauce, sprinkle with cilantro, and finish with a dab of sour cream.