Preheat the oven to 400 degrees F. Using a meat mallet or heavy rolling pin, pound the chicken breasts a bit, just to even out the thickness.*
Heat the olive oil in a large oven-safe pan (cast iron is ideal) over medium-high heat.
Sprinkle the flour on a plate or shallow dish. Season the chicken breasts with salt and pepper and dredge in the flour, shaking off the excess.
Add chicken breasts to the hot pan and sear for about 2 minutes on each side, until golden brown.
While the chicken is searing, stir together the hot sauce and melted butter. Pour sauce over the chicken breasts and transfer the pan to the oven. (Note: if your pan is not oven safe, transfer the chicken breasts to a baking dish before pouring on the sauce and placing in the oven)
Bake chicken breasts for 20-25 minutes, until just cooked through. Add cheese slices to the tops and return to the oven to melt for a minute or two.
Top each burger bun with a chicken breast, ranch or blue cheese dressing, and additional toppings of your choice.
*After I've pounded the chicken breasts a bit, I also like to slice off the chicken tenders (small dangly part) and trim them to a roundish shape to fit the buns better, but this is totally optional. I freeze the trimmings to use in soups and such.