Mix the feta, yogurt, and lemon juice with a fork, crushing the feta.
Add chopped fresh cilantro, chopped fresh mint, and minced garlic, and mix until combined.
To the ground lamb, add chopped mint, chopped cilantro, minced garlic, Dijon mustard, salt, ground black pepper, lemon juice, and lemon zest. Mix until everything is combined.
Shape 2 patties from the lamb mixture.
To a pan, over medium-high heat, add 1 tbsp of oil. When the oil is hot, add the lamb patties and cook for about 5 minutes on each side until fully cooked (at a minimum, lamb patties should be cooked to 160°F (71.1° C)).
While the lamb patties are cooking, prepare your toppings: slice the tomatoes, peel and slice the red onion, and cut thin slices of cucumber (using a vegetable peeler or a pairing knife).
(Optional) Toast the buns in a pan over medium heat for about 1-2 minutes.
Assemble the burgers: toasted bun, feta-yogurt mixture, lamb patty, tomato, onion, and cucumber.