Preheat the oven to 200°C/fan 180°C/gas mark 6. Trim the beetroot stalks so there is only about an inch left attached to the bulb, then wrap in tin foil and place on a small baking tray. Transfer to the oven and bake for 1 hour to 1 hour 15 minutes. When the beetroot is cooked, remove from the oven, open up the tin foil and leave to cool for 15 minutes.
Meanwhile, place the carrot in a heatproof bowl with the turmeric and 1 tbsp water, then cover with cling film and make a small slit in the centre to release steam. Place in the microwave and cook on a high heat for 5 minutes, or until completely tender. Once the carrot is cooked, place in the small bowl of a food processor with a splash of water and blitz until smooth. Transfer to a bowl.
Meanwhile, place the milk in a heatproof mug and microwave on a high heat for 20 seconds until warm. Add the flour to the bowl of a standing mixer with the dough hook attached. Add the salt and sugar on one side and the dried yeast on the other. Carefully pour the warm milk, eggs and carrot purée into the middle and mix on a slow speed for 2 minutes, then increase the speed to medium for 5 minutes. Gradually add the butter to the dough, mixing for 5 more minutes, and scrape down the sides to ensure the butter is mixed in. The dough should feel soft, but not too sticky. Lightly grease a mixing bowl. Add the carrot dough, cover with greased cling film and leave in a warmish place to prove for an hour, or until doubled in size.
Peel the stalks and skin off the beetroot, then quarter the bulbs. Transfer them to the cleaned small bowl of the food processor and blitz until smooth. Strain the beetroot purée with a spoon against the side of the bowl. Repeat step 3, but with the beetroot purée instead of the carrot.
Lightly oil a 1.8kg loaf tin (23x11x8cm) and preheat the oven to 220°C/fan 200°C/gas mark 6. Turn out the carrot dough on to a clean surface and knock out the dough by punching it with your fist. Divide the dough into two and loosely roll into two sausage shapes the length of the loaf tin. Repeat with the beetroot dough.
Place a sausage of the carrot dough neatly beside a sausage of the beetroot dough in the base of the tin, then layer the 2 leftover sausages on top, making sure the second carrot sausage sits on top of the first beetroot sausage, to make a checkerboard effect. Cover again with lightly oiled cling film and leave to rest for a final 30 minutes, until slightly puffed up.
Brush the unbaked loaf with the beaten egg, sprinkle over the caraway seeds and transfer to the oven to bake for 15 minutes. Lower the oven temperature to 190°C/fan 170°C/gas mark 5 for a further 25 minutes.
Once baked, remove from the oven and transfer to a cooling rack for 10 minutes. Remove the loaf from the tin and cool for a further 20 minutes. You may need to gently ease the bread out by sliding a sharp knife around the edge of the tin.