Mix ground beef, feta cheese, goat cheese, 2 tablespoons lemon juice, olive oil, oregano, and garlic together in a large bowl using your hands. Form meat into 4 patties.
Stir Greek yogurt, cucumber, and 3 tablespoons lemon juice together in a bowl.
Heat a frying pan over medium heat for about 2 minutes; cook burgers to desired degree of doneness, about 5 minutes per side. An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C).