Peel, wash and slice the carrots. Clean, wash and cut the peppers into pieces. Wash potatoes thoroughly and cut in half. Pour artichokes into a sieve, drain and halve.
Wash the thyme, shake dry. Pluck the leaves from 3 stems. Wash rosemary, shake dry and pluck needles from the twigs. Mix vegetables, potatoes, rosemary, thyme stems and leaves, salt and pepper.
Place on a baking tray and sprinkle with 3 tablespoons of oil. Cook in the preheated oven (electric cooker: 175 °C/ circulating air: 150 °C/ gas: level 2) for approx. 50 minutes. In the meantime, peel the garlic and press it through a garlic press.
Put the garlic and egg yolk into a high mixing bowl and mix with the whisks of the hand mixer. Add 125 ml of oil drop by drop while stirring continuously. Stir in yoghurt. Season with salt and pepper.
Cut aioli into thirds. Wash basil, shake dry, put 2-3 leaves aside for garnishing. Cut the rest of the basil into fine strips and stir into 1/3 of the aioli. Chop tomatoes very finely. Add the tomatoes and capers to the 2nd aioli.
Stir aioli for a third. Pour the aioli into 3 bowls, garnish basil aioli with basil leaves. Pass the aioli to the oven vegetables.