Grate cucumber and squeeze dry using a kitchen towel.
Mix yogurt, cucumber, garlic, dill, lemon juice, olive oil, salt, and pepper. Refrigerate.
Combine ground lamb, feta, garlic, herbs, spices, egg, salt, and pepper in a bowl.
Shape into 4–6 patties and refrigerate for 10 minutes.
Grill or pan-fry patties 4–5 minutes per side over medium-high heat.
Toast buns or warm pita while burgers rest.
Spread tzatziki on bun, add lettuce, patty, toppings, and more sauce.