Begin by building the base of the patties. Grate the red onion directly into a large bowl, followed by the garlic worked into a fine paste. Add the ground lamb, crumbled feta, oregano, cumin, salt, and black pepper. Using your hands, fold everything together gently — just enough to bring it into cohesion. Overworking at this stage will tighten the texture, and what you want is something tender, almost loose, that yields beautifully once cooked.
In a separate bowl, prepare the mint yoghurt. Stir together the Greek yoghurt, finely sliced mint, lemon juice, zest, and the grated cucumber. Take a moment to press the cucumber dry beforehand — it is a small detail, but it keeps the sauce thick, cool, and luxuriously spoonable rather than diluted.
Divide the lamb mixture into four equal portions and shape into patties. Press a shallow indentation into the centre of each — a simple touch that ensures they remain flat and evenly cooked, rather than swelling as they hit the heat.
Place the patties onto a preheated grill set to medium-high. Let them cook undisturbed for 4 to 5 minutes, allowing a deep, caramelised crust to form. Turn once, and cook for a further 4 minutes until richly coloured and just cooked through. Resist the urge to press them down — that pressure forces out the very juices you have worked to preserve.
While the patties finish, lay the brioche buns cut-side down onto the grill. In under a minute, they will take on a light golden edge and release that unmistakable buttery aroma.
Transfer the lamb to a plate and allow it to rest briefly — just five minutes is enough for the juices to settle and the texture to relax.
To assemble, layer the base of each bun with a handful of greens, followed by a thick slice of tomato. Set the lamb patty on top, then finish with a generous spooning of the mint yoghurt. Crown with the top bun and serve immediately, while everything is still warm, cool, and perfectly in balance.