Peel, seed, grate and squeeze out all the water from the cucumber.
Add the Greek yogurt, ¼ cup diced red onion, 1 teaspoon dried oregano, prepared cucumber, 1 minced garlic, and 1 teaspoon lemon zest in a medium bowl and mix well.
Chill until needed.
To make the lamb burgers:
In a large bowl, combine the ground lamb, red onion, crumbled feta, lemon zest, oregano, garlic, salt, and pepper. Mix well until thoroughtly combined. .
Form the meat mixture into 6 even sized patties and cook over medium-high heat for 4-5 minutes per side, or until your desired doneness.
Remove the burgers from the heat and allow to sit (covered) for 10 minutes before serving.
Serve on buns along with the chilled tzatziki sauce, sliced tomatoes, and lettuce if desired.