Make the Tzatziki Sauce: Place the grated cucumber into a clean kitchen towel or a piece of cheesecloth. Twist and squeeze tightly over the sink to wring out as much excess liquid as possible. In a medium mixing bowl, whisk together the Greek yogurt, drained cucumber, minced garlic, lemon juice, fresh dill, salt, and black pepper until well blended. Cover the bowl and chill in the refrigerator for at least 30 minutes before serving.
Mix the Burger Ingredients: Grab a large bowl. Toss in your ground chicken, crumbled feta, chopped spinach, minced red onion, minced garlic, dried oregano, lemon zest, and lemon juice. Add a pinch of salt and pepper. Give it a gentle mix until everything is just combined.
Form the Patties: Take the mixture and form it into 4 equal patties. I try to make them the same size so they cook evenly, and I like to press a tiny dimple in the center so they stay flat.
Cook to Perfection: Heat your olive oil in a pan or on the grill over medium heat. Let those patties cook for about 5 to 7 minutes on each side. You want them fully cooked through (reaching an internal temperature of 165°F) and nice and golden.
Assemble and Serve: Toast your brioche buns. Slather a generous spoonful of your chilled tzatziki sauce right onto the bun. Add your burger, lettuce, tomato, crisp cucumber, and red onion. Now, enjoy every single bite!