Combine mint, vinegar, sugar, mustard, and salt in a food processor. Pulse until mint is finely chopped. With motor running, add boiling water and process until sauce is bright green and pourable, about 30 seconds. Transfer to a bowl and let sit at room temperature for at least 15 minutes so flavors meld.

Pat lamb chops completely dry with paper towels - this is crucial for good browning. Rub all over with olive oil, then season generously with salt and pepper. Let stand at room temperature while the grill heats.

Heat grill to high (about 500 degrees F) or heat a grill pan over high until smoking. Place chops on grill and cook without moving until deep brown char marks form, 3 to 4 minutes. Flip and continue cooking until an instant-read thermometer inserted into the thickest part registers 130 degrees F for medium-rare, about 3 minutes more. Transfer to a plate and let rest for 5 minutes.

Arrange chops on a platter or individual plates. Spoon mint sauce over the top or serve alongside for dipping. The sauce will keep, covered and refrigerated, for up to 3 days.