Peel and chop the onion. Peel the garlic cloves and crush or finely grate. Give the carrot a scrub and dice it into small pieces.
Pour 1 tbsp oil into a pan and bring to a medium heat. Slide in the onion, garlic and carrot. Sprinkle in a generous pinch of salt and pepper and fry for 5 mins, stirring occasionally, till slightly softened.
While the onions are cooking, trim the beetroot and coarsely grate. When the vegetables in the pan are softened, tip in the beetroot. Pick the tarragon leaves and sprinkle in half (save the rest for later). Top up with 750ml boiling water and crumble in the stock cube. Stir well combine and bring to the boil, then reduce to a simmer and cook for 20 mins.
Finely grate the orange zest and set aside. Squeeze the juice into the pan. Use a stick blender to whizz into a smooth soup. No stick blender? Carefully pour the contents of the pan (in batches if necessary) into a food processor and blend.
Spoon the soup into bowls and finish with the orange zest and the remaining tarragon leaves.