In a bowl, combine the minced beef with the grated red onion, salt and pepper. Mix gently, then divide into 4–6 portions and shape into slightly flattened patties.
Heat 2 Tbsps of olive oil in a frying pan over medium-high heat. Once hot, cook the patties for about 4–5 minutes on each side, until nicely browned and cooked through. Keep warm.
Lightly oil a grill pan and cook the pineapple slices on both sides until golden and lightly caramelised. Toast the halved burger buns in the same pan. In the remaining oil, fry the eggs sunny-side up.
Assemble the burgers in this order: bottom bun, grilled pineapple, a lettuce leaf, 2–3 slices of beetroot, a slice of tomato, the beef patty, then the fried egg. Finish with the top bun.
Serve immediately while warm - on its own, or with fries and a crisp green salad.