Mix together three-quarters of the shallot, the egg yolk and beef and season well. Wet your hands and shape the mixture into 4 burgers. Brush all over with a little oil, place on a baking sheet and cook under a preheated hot grill for 5 minutes until golden.
Turn the burgers over, arrange the polenta chips on the baking sheet and drizzle with a little more oil. Cook for a further 3 minutes, then turn over the polenta chips and place the mozzarella on top of the burgers. Return to the grill for 1–2 minutes until the cheese has melted and the polenta is golden.
Meanwhile, mix together the tomatoes, the remaining shallot and oil, the balsamic vinegar and basil and season to taste. Place the burgers in the buns with a handful of rocket and serve with the tomato salad and polenta chips.