In a small bowl, combine the grated cucumber, Greek yogurt, lemon juice, and a pinch of salt to make the quick tzatziki sauce. Stir well until smooth, then set aside in the refrigerator to let the flavors meld.

In a large bowl, gently mix the ground lamb, crumbled feta cheese, minced garlic, chopped oregano, kosher salt, and black pepper. Mix just until combined to ensure the burgers stay tender; do not overwork the meat.

Divide the lamb mixture into four equal portions and shape them into round patties about 1/2-inch thick. Use your thumb to make a shallow indentation in the center of each patty, which will prevent them from puffing up while cooking.

Heat a large skillet or grill pan over medium-high heat. Once hot, add the lamb patties and cook for 4 to 5 minutes per side, or until a deep brown crust forms and the internal temperature reaches 160°F for medium-well.

Assemble the burgers by spreading a generous dollop of the chilled tzatziki sauce on the bottom half of each toasted bun. Top with fresh spinach, the hot lamb patty, sliced red onion, and the top bun, then serve immediately with crispy fries or a side salad.