Preheat the oven to 450°F. Pat the rack of lamb dry, trim any thick exterior fat to about 0.25 inch, and score the fat cap in a shallow crosshatch.
Season the lamb all over with the kosher salt and black pepper.
Heat the vegetable oil in an ovenproof skillet over medium-high heat until shimmering. Sear the lamb, fat side down first, until well browned and some fat is rendered, 3–4 minutes; turn and sear the other side for 2 minutes. Transfer to a plate briefly.
In a bowl, combine the fresh breadcrumbs, fresh parsley, fresh rosemary, fresh thyme, garlic, and olive oil until evenly moistened and crumbly.
Brush the seared lamb (fat side) with the Dijon mustard in a thin, even layer, then press the breadcrumb mixture onto the mustard-coated side to form a compact crust.
Set the lamb, crust side up, back in the skillet or on a rimmed baking sheet with a rack. Roast until the center registers 120–125°F for medium-rare, 15–20 minutes, checking with an instant-read thermometer inserted between bones into the thickest part.
Transfer to a cutting board and rest until juices settle and temperature rises to 130–135°F, about 10 minutes.
Slice between the bones into chops and serve immediately.