Pre-heat the oven to Gas Mark 6, 200°C, fan 180°C. Slice the potatoes as thinly as possible into mandolin and pat dry. In a medium saucepan, heat the milk, thyme and garlic over a medium heat to boiling point.
Whisk in the nutmeg and set to one side to cool a little. Strain into a jug once cooled. Meanwhile, blanch the broccoli in boiling salted water for 1 minute. Remove and refresh in ice water.
Layer the slices of potato in a 7 inch metal cake tin, lined with nonstick baking paper. Arrange a few broccoli pieces on top and cover with some of the milk mixture, seasoning well with salt and pepper in between layers. Layer the rest of the potatoes, adding broccoli in between and adding more of the milk to cover. Finish the Dauphinoise with a layer of potato.
Cover the cake tin with tin foil and arrange on a baking sheet. Bake for 45 minutes to 1 hour and then remove the foil. Bake for a further 10 minutes until the top is turning golden brown and the potato is soft.
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