Set your oven to 400°F (200°C). Lightly grease a 9×13-inch (23x33cm) baking dish with olive oil or nonstick spray.
Grate the cucumber using a box grater. Place it in a clean kitchen towel or several layers of paper towel and squeeze firmly until almost no water comes out. This step matters more than it sounds. Wet cucumber makes the sauce runny and prevents the golden crust from forming.
In a medium bowl, combine the Greek yogurt, squeezed cucumber, minced garlic, lemon juice, olive oil, dill, mint (if using), salt, and pepper. Stir until smooth and well combined. Taste it. The sauce should be tangy, garlicky, and bright. Adjust lemon or salt if needed.
Pat the chicken pieces dry with paper towels. This helps the sauce stick better and promotes browning. Place the chicken in the prepared baking dish. Drizzle with 1 tablespoon olive oil and season with ½ teaspoon salt and ¼ teaspoon black pepper. If using optional vegetables, tuck them around the chicken pieces now.
Spoon the tzatziki sauce over each piece of chicken. Use the back of a spoon to spread it into a thick, even layer that fully covers the top and sides. Don’t be shy here. The more sauce, the better the results.
Place the dish uncovered in the preheated oven. Bake for 38–40 minutes for chicken breasts, or 35–38 minutes for thighs. The chicken is done when the internal temperature reads 165°F (74°C) on an instant-read thermometer and the edges of the sauce turn slightly golden with a few darker, charred spots on top. Those spots are flavor, not burning.
Remove from the oven. Scatter crumbled feta over the top immediately so it softens slightly from the residual heat. Let the dish rest for 5 full minutes before serving. This short rest allows the juices to redistribute back into the meat.
Finish with fresh dill or parsley and a few lemon slices on the side. Serve directly from the baking dish.