Put the cutlets in a deep baking tray along with the onion, thyme, garlic, cayenne (adjust depending on how spicy you’d like the chops to be) and lemon juice. Drizzle with 1 tsp of oil and season well, tossing together. Cover with foil and leave to marinate for at least 4 hrs, ideally overnight.
Heat the oven to 140C/120C fan/gas 1. Remove the tray from the fridge 30 mins before cooking. Add a splash of water to the tray, cover again tightly with foil, and cook for 15 mins until the cutlets are just cooked through and tender. Discard the contents of the tray, keeping the lamb.
Put the remaining oil into a frying pan over a high heat. Sear four of the cutlets for 1 min-1 min 30 seconds on each side without moving, including the side of fat, until deeply golden brown. Repeat with the remaining cutlets. Transfer to a plate and leave to rest.
While the cutlets rest, start on the sauce. Put the oil and butter in a pan over a low-medium heat. Once melted, cook the shallot for 5-6 mins or until softened. Remove from the heat and add the mustards, lemon juice and sugar. Strain in the juices left from the rested lamb along with 1 tbsp of water and whisk together, lightly seasoning to taste. Serve with the chops garnished with the thyme.