Mince the garlic cloves finely and finely chop the spinach into thin ribbons. Dice the red onion into small pieces and zest the lemon directly into a small bowl. Crumble the feta cheese into small pea-sized pieces, keeping them separate from the other ingredients. Measure out the breadcrumbs and set aside. Arrange all prepped ingredients near your work surface for efficient assembly.
In a large bowl, add the ground chicken and crack both eggs directly into it. Sprinkle the breadcrumbs, oregano, dill, salt, and pepper over the top. Add the minced garlic, lemon zest, and finely diced red onion. Gently fold everything together with your hands until just combined—avoid overworking the mixture, as this can make the patties dense. Fold in the chopped spinach and crumbled feta last, being careful not to break up the cheese too much. Divide into 5 equal portions and shape each into a 1-inch thick patty. I like to make a small indentation in the center of each patty with my thumb—this helps them cook evenly and prevents them from puffing up during cooking.
Heat a large skillet or griddle over medium heat for about 2 minutes until hot but not smoking. Carefully place the patties into the pan, leaving space between each one. Cook for 4-6 minutes on the first side without moving them—this allows a light crust to form and the feta to hold the patty together. Flip gently and cook for another 4-6 minutes on the second side. Check the internal temperature with an instant-read thermometer; it should reach 165°F when inserted into the thickest part of a patty. Remove the burgers from heat and let them rest on a plate for 2 minutes before assembling.
While the burgers cook, slice the red onion into very thin half-moons and place in a small bowl. Slice the cucumber into rounds or spears and chop the tomato into slices or wedges. Toast the whole wheat buns until golden brown and crispy—this adds structural support and prevents them from falling apart when you add the wet toppings. I find that a toasted bun makes all the difference with these juicy burgers because the breadcrumbs can release moisture during cooking.
Place each cooked burger patty on the bottom half of a toasted bun. Spread about 2 tablespoons of tzatziki on the top half of the bun. Layer the fresh cucumber slices, tomato, and thin red onion slices on top of the burger. Close the bun and serve immediately while the patty is still warm and the vegetables are fresh and crisp.