Preheat the oven to 200C, fan-forced. Thinly slice the cheese and tomato. Peel and coarsely grate the beetroot.
Step 2 — Make beetroot relish
Heat 2 tsp olive oil in a small saucepan over medium heat. Cook the beetroot, stirring, for 5 mins or until softened. Add 1 tbs water, 2 tsp sugar and 2 tsp balsamic vinegar, reduce the heat to medium-low and cook, stirring occasionally, for 5 mins or until slightly sticky. Remove the pan from the heat. Season with salt and pepper.
Step 3 — Brown beef patties
While the beetroot relish is cooking, heat 1 tsp olive oil in a medium frypan over medium-high heat. Add the beef patties, season with salt and pepper, then cover and cook for 3 mins each side or until browned (the patties won't be cooked through). Remove the patties from the pan and drain excess oil.
Step 4 — Finish patties
Add half the Aussie barbecue seasoning*, 1 tbs water and ½ tsp sugar to the pan and stir to combine. Return the patties to the pan and cook, turning halfway, for 1-2 mins until cooked through and the liquid is reduced. Put the cheese onto the patties and cook, covered, for a further 1 min or until the cheese is melted. Remove the pan from the heat and keep covered.
Step 5 — Toast rolls and fry eggs
Halve the rolls through the middle. Put on an oven tray, cut-side up, and bake for 6 mins or until lightly toasted. Heat 1 tsp olive oil in a separate medium frypan over medium heat. Crack 2 eggs into the pan and cook for 3-4 mins until just set, or until cooked to your liking.
Step 6 — Assemble and serve
Spread the roll bases with some of the beetroot relish. Fill with the tomato, patties, egg and some of the salad leaves, then sandwich with the roll tops. Drizzle the remaining salad leaves with a little extra virgin olive oil, if desired, and season with salt and pepper. Serve the burgers with the salad leaves and remaining beetroot relish.