Preheat oven to 350 degrees Fahrenheit. Heat a dry 12-inch heavy bottom pan (or oven safe pan) over high heat until hot, at least 2 minutes.
Meanwhile, pat lamb dry and rub meat all over with salt and pepper.
Add oil to hot skillet, then brown racks, one at a time.
Remove racks from the pan and wrap the bones in aluminum foil.
Wipe skillet out and return the racks to the skillet with the bones facing up and the meat facing towards the outside of the pan.
Place pan in the oven and roast 15-20 minutes or until thermometer inserted diagonally into center of meat registers 120°F.
Remove the lamb from the oven and transfer to a plate. Let stand, loosely covered in foil for 10 minutes.
Pour out all but 1 tablespoon of collected fat from the pan.
Add in the apple pieces, brown sugar, salt, black pepper, and half the butter. Allow the apples to roast until golden brown and caramelized.
Deglaze the pan with the cider, allowing it to reduce by half, and add the chopped mint.
Remove from the heat and add the remaining butter.
Swirl to combine and adjust the seasoning with salt if necessary.
Slice into chops and serve with the apples and sauce.