Heat oil in small saucepan; deep-fry thyme briefly, only about 5 seconds til fragrant. Remove thyme from oila and drain on absorbent paper.
Prehead oven to moderate.
Cut potatoes into 1cm slices; heat 2 tablespoons of the reserved thyme oil in ovenproof/flameproof baking dish; cook potato slices in batches til lightly browned on both sides. Return all potatoes to baking dish; add lamb and roast, uncovered for about 15 minutes or unil cooked as desired.
Meanwhile, make aioli.
Boil, steam or microwave spinach til just wilted; drain. Heat remaining thyme oil with butter in small saucepan; cook mushrooms and garlic til softened. Cut each lamb rack into 4 cutlets; divide among serving plates with spinach, potato and mushrooms. Top with aioli, garnish with fried thyme and drizzle with vinegar.
Aioli: whisk mustard, vinegar, garlic and egg yolks in a small bowl til combined; gradually add oil in a thin, steady stream, whisking constantly, til aioli thickens. Whisk in lemon juice.