For the crust: Preheat the oven to 400 degrees F (200 degrees C). Line a large baking sheet with foil. Prepare a medium bowl of ice water and set aside.

Bring a small pot of water to a boil. Boil mint leaves for about 10 seconds. Transfer mint leaves to the reserved ice water for 30 seconds. Drain and squeeze dry.

Combine mint with garlic, olive oil, bread crumbs, salt, black pepper, and cayenne pepper in a food processor or blender. Blend for 20 to 30 seconds until coarsely chopped. Transfer mint mixture to a medium bowl. Stir in Parmigiano-Reggiano cheese and set aside.

Combine Dijon mustard and honey in a small bowl and set aside.

Slice 1/2 to 1-inch slits between each rib bone to ensure the meat roasts evenly. Season each rack with salt and black pepper on all sides. Heat vegetable oil in a large skillet over high heat. Brown each lamb rack on meat side, 2 to 3 minutes. Transfer lamb racks to the foil-lined baking sheet, meat side up.

Brush each rack with reserved mustard-honey mixture. Sprinkle mint-bread crumb mixture evenly over top and sides of each rack, pressing lightly into meat so mixture adheres. Roast in the preheated oven until browned on top and medium-rare in the center, about 20 minutes. An instant-read thermometer inserted into the center should read 125 to 130 degrees F (52 to 54 degrees C).

Make the vinaigrette: Combine extra-virgin olive oil, rice vinegar, honey, Dijon mustard, salt, and black pepper in a jar with a lid. Shake vigorously until emulsified, about 30 seconds.

Just before serving, drizzle the vinaigrette evenly over the lamb.