Heat the oven to 200°C/400°F. Spread the beetroot on a baking sheet, drizzle with oil and sprinkle on salt and pepper and roast for 30 minutes.
Once the beetroot has finished cooking leave to cool for a couple of minutes then place in a food processor and pulse a few times until chopped but still a little chunky. Put the beetroot in a large bowl and add the bulgur wheat, egg, garlic, breadcrumbs, oil, lemon juice, chilli flakes and salt and pepper. Mix until completely combined.
Shape the mixture into four patties and place on a lined baking sheet. Place in the oven and cook for 30 minutes, flipping halfway through.
While the burgers are cooking make the whipped feta. Put the yoghurt and feta in a blender and mix until smooth, add the herbs and pepper and blend again until smooth. Serve the burgers in brioche buns topped with the whipped feta and rocket leaves.
Nutrition information is an estimate and will vary depending on the exact ingredients used.