Start by making the tzaziki sauce. Grate half of the cucumber. Place the grated cucumber in a coffee filter or cheesecloth and squeeze out the excess liquid. Add the squeezed grated cucumber to a medium bowl. Add all other tzaziki sauce ingredients, and mix well to combine. Refrigerate for 1-2 hours or covered overnight.
Prepare the chopped vegetable topping for the gyro tacos. Add diced onion, diced cucumber, diced kalamata olives, diced tomatoes, juice of half lemon, half tablespoon of olive oil, and about half teaspoon of Greek seasoning to a medium bowl. Mix all ingredients together and refrigerate until ready to serve.
Heat griddle to about 375-400 degrees F.
Make the ground lamb balls. Add one pound of ground lamb to a large bowl with all of the herbs, spices, and breadcrumbs listed in the "For the Lamb" ingredient list. Use your hands to mix together well. Portion the meat mixture out into 6 equally sized balls. *You can use a 1/3 cup measuring cup to equally portion the meat.
When griddle is hot, place one lamb ball on the griddle surface. Cover the meat ball with one piece of pita bread, and smash the lamb into a patty underneath the pita. Apply pressure to the pita and lamb patty for about 6-8 seconds. Repeat this process for the other 5 lamb balls.
Cook the Greek lamb smashed tacos on the first side for about 3 minutes. Then, flip to the second side (pita side down) and cook an additional 2-3 minutes, or until the lamb patty reaches at least 160 degrees F internal temperature. Then, remove the smashed gyro tacos from the griddle.
Top each gyro taco with tzaziki sauce, shredded lettuce, some of the chopped Mediterranean "pico" vegetable mix, and crumbled feta cheese. Serve warm.