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Make the aioli the day before; chill. Roast the potatoes an hour ahead. Note: this recipe contains raw/partially cooked egg.
Preheat the oven to 200°C, fan 180°C, gas 6. Wash and drain the potatoes, toss with 1 teaspoon sea salt while still damp. Tip on to a baking sheet and roast for 1 hour until tender.
Using an electric hand whisk, whisk the crushed garlic and mustard powder into the egg yolk. Add the oil, whisking constantly, a drop at a time at first, increasing to a slow, steady stream as the aioli thickens. When the oil is incorporated, season to taste with salt, pepper and lemon juice. Spoon into a serving bowl or glasses.
Peel the carrots and trim the green tops. Separate the chicory leaves. Arrange the radishes, carrots and chicory leaves on a large serving platter or in the glasses with the roasted new potatoes and aïoli.