Melt the butter in a skillet over medium heat.
Sprinkle the flour over the melted butter and whisk until foamy and smooth, about 1 minute.
Slowly pour in the milk while whisking constantly, and cook until the sauce thickens and coats the back of a spoon.
Remove the skillet from the heat and stir in 1 cup shredded Monterey Jack cheese until melted.
Stir in the salt, garlic powder, onion powder, cumin, and cayenne until fully incorporated; set the queso aside and keep warm.
Arrange the tater tots in a single layer on a baking sheet or in a 9x13-inch pan and bake according to package directions until golden and crisp.
While the tots bake, combine the shredded chicken and barbecue sauce in a bowl until the chicken is evenly coated.
Spoon some queso over the baked tater tots, then top with the BBQ chicken and sprinkle with the additional shredded cheese.
Add diced tomatoes, sliced jalapeños, and diced red onion over the top as desired.
Return the pan to the oven and bake 7–9 minutes more, or until everything is warmed through and the cheese is melted.
Remove from the oven, dollop with sour cream if using, and serve immediately.