1.Heat oil in a small saucepan; deep-fry thyme, about 5 seconds or until fragrant. Drain on absorbent paper and reserve oil.
2.Preheat oven to 200°C.
3.Cut potatoes into 1cm slices. Heat 2 tbsp of the reserved oil in a flameproof baking dish. In batches, cook potato slices until lightly browned on both sides. Return potato to dish.
4.Add lamb to dish and roast about 20 minutes or until cooked as desired. Cover to keep warm.
5.Meanwhile, to make aïoli, whisk mustard, vinegar, garlic and egg yolks in a small bowl until combined. Gradually add oil in a thin, steady stream, whisking constantly, until mixture thickens. Whisk in juice.
6.Boil, steam or microwave spinach until wilted; drain.
7.Heat remaining reserved oil with butter in a small saucepan. Cook mushrooms and garlic, stirring, until mushrooms soften.
8.Cut each lamb rack into four cutlets. Divide among serving plates with spinach, potato and mushrooms. Top with aïoli and fried thyme; drizzle with vinegar.