Preheat the oven to fan 160C/ conventional 180C/gas 4. Heat 1 tbsp of the olive oil in a frying pan, add the beef and quickly fry on all sides until nicely browned. Transfer to a shallow ovenproof dish and season all over with salt and pepper. (Don’t wash the pan; you can use it again.) Roast for 35-45 minutes, depending on how well you like your meat cooked and the thickness of the joint. Leave to cool.
Meanwhile, peel the shallots, leaving the roots intact, and cut in half lengthways. Heat another 1 tbsp olive oil in the frying pan, add the shallots and gently cook for 10 minutes, turning until softened. Sprinkle with the caster sugar and cook for a further few minutes, then remove from the heat and tip into a large bowl. Add the vinegar and remaining 3 tbsp olive oil, season with salt and pepper and leave to cool.
Heat vegetable oil, about 2.5cm deep, in a wide pan or wok. Thinly slice the beetroot and spread out the slices on a tray lined with kitchen paper. Pat them dry with more kitchen paper. Check the oil is hot enough by carefully dropping in a slice of beetroot – if it immediately rises to the surface covered with bubbles, the oil is ready. Fry the beetroot slices in batches for 2-3 minutes until they are curled and crisp. Drain well on a large plate lined with kitchen paper and leave to cool.
To make the dressing, mix all the ingredients in a small bowl and season to taste. Cut the cold beef into thin slices using a very sharp, unserrated knife. Put the salad leaves in the bowl with the shallots, toss well and spread over a large serving platter. Arrange the slices of beef on top of the leaves, spooning a little dressing on each slice as you go. Scatter the beetroot crisps over the dish and serve with oven-sautéed new potatoes (see below).