Heat the oil in a 12-inch stainless steel skillet over medium-high heat until shimmering. Lay the lamb racks in the pan, fat side down, and cook, turning occassionally, and basting regularly with the browned butter (tobaste, tilt the pan toward you so that the butter pools at the bottom and spoon the butter over the lamb, taking care to hit any underbrowned spots repeatedly), until the lamb is deep golden brown and an instant-read thermometer inserted into the thickest part of each rack section registers 120F for medium-rare, or 130F for medium, 3 to 7 minutes longer. Transfer the lamb racks to a rack set on a rimmed baking sheet and let rest for 5 minutes.