Place the potatoes on one baking sheet and use a fork to poke holes all over the potatoes.
Brush or rub the olive oil over the potatoes.
Place the chicken on the other baking sheet and brush with ¼ cup of the barbecue sauce.
Place the baking sheets with the chicken and potatoes in the preheated oven. Bake the chicken for 25–35 minutes, until the internal temperature reaches 165 F. Set aside until the potatoes are done.
Bake the potatoes until tender, about 1 hour, depending on the size. Set the potatoes aside until cool enough to handle.
Use two forks to shred the cooked chicken.
Add the shredded chicken to a medium bowl, add the remaining ¼ cup barbecue sauce, and stir to coat.
Slice the potatoes in half lengthwise.
Scoop the insides of the potatoes into a large bowl, and set the skins aside.
Add the sour cream, melted butter, salt, pepper, half of the cheddar, and half of the Monterey jack to the bowl with the potatoes. Mix with a hand mixer until combined and creamy.
Spoon the potato mixture evenly into the potato skins.
Top each potato with the shredded barbecue chicken.
Add the remaining cheese on top of the chicken.
Return the potatoes to the preheated oven and bake until the cheese is melted, 10–12 minutes.
Serve hot, topped with sliced green onions, if desired.