Rinse and drain the millet and place into a small saucepan with the salted water. Place over a medium heat, bring to a simmer, cover and cook over a very low heat for 20 to 25 minutes until tender. Remove from the heat and leave to stand for 5 minutes, covered.
Put the beetroot, courgette, carrots and mint into a large bowl. Add the millet, vinegar, oil, a pinch of salt and pepper then mix well. Add the cornflour and egg and mix again, then chill in the refrigerator for 2 hours.
Put the yoghurt in a fine strainer over a bowl and drain for at least 40 minutes. Stir in the garlic and season to taste with salt and pepper.
Pack the beetroot mixture into a 125ml cup, then turn out and shape into a patty. Repeat to make five burgers. Place a ridged griddle pan or large frying pan over a medium heat and coat with oil. Add the patties and cook for approximately 5 minutes on each side, turning carefully, until brown.
Spread the buns with the yoghurt sauce and place the burgers in the buns, topped with lettuce, a slice of beef tomato, and sliced pickled gherkin. Serve immediately.