Put the ground lamb, pork, or chicken in a bowl. Add the oregano, paprika, kosher salt, pepper, crushed red pepper flakes, and garlic. Mix together with your hands. Roll the ground meat into 8 to 10 balls.
In another bowl add the Greek yogurt, mayo, grated cucumber, lemon juice, kosher salt, pepper, and crushed red pepper flakes. Stir until combined.
Preheat the griddle to medium heat for a few minutes. Put the balls of meat on the griddle spaced apart, put a flour tortilla on top of each ball. Use a burger press to smash each tortilla over the meat, pressing as flat as possible.
Cook for 3 to 4 minutes, flip and cook another minute or two.
Top the tacos with the sauce from earlier, shredded lettuce, and crumbled feta cheese.