With a sharp knife, score the fat side of the lamb in a crosshatch pattern. Generously season all over with salt and pepper. Set aside and let come to room temperature for 1 hour before cooking.
Preheat oven to 375F.
Fill a sauce pan with water, add the potatoes and set over high heat. Cook until slightly fork-tender, around 10 minutes total. Drain, let the potatoes cool to the touch, and slice each of them in half, lengthwise. Set aside.
In a food processor, combine the parsley, garlic, rosemary and thyme. Pulse until finely chopped. Add the crackers and pulse until well combined. Transfer the mixture to a board or plate and spread into an even layer. Set aside.
Heat a large, oven-safe pan over medium heat for 5 minutes. Add the avocado oil and heat until shimmering. Sear the lamb, fat-side down at first, until golden brown, around 4 minutes. Flip to sear the bottom side until golden brown, 4 more minutes. Transfer to a plate and set aside.
Drain all but 2 tablespoons of fat in the pan. Add the potatoes, carrots and shallots and season with a pinch of salt and pepper. Toss to coat and cook 2 minutes. Remove from heat and set aside.
Brush the lamb with mustard. Using your hands, evenly coat the meat in the herb and garlic mixture. Tent the bones with aluminum foil to prevent burning, and lay the rack of lamb, meat side-up, on top of the vegetables. Transfer to the oven and cook until the thickest part of the lamb registers 125F on an instant read thermometer for medium-rare, around 18 minutes.
Transfer the lamb to a cutting board and let it rest, 10 minutes. Return the vegetables to the oven and continue cooking until golden brown and tender, 10 minutes.
Carve the lamb and serve with the vegetables.