Make the yogurt sauce. Add the Greek yogurt, olive oil, garlic, lemon juice, cumin, salt, black pepper, and optional cayenne to a small bowl. Stir until smooth and well combined. Refrigerate while preparing the kofta.
Prepare the kofta mixture. Place the ground beef, za’atar, salt, black pepper, shallot, parsley or cilantro, garlic, and beaten egg in a large bowl.
Mix gently. Use clean hands or a fork to combine the ingredients just until evenly incorporated. Avoid overworking the beef, which can make the cooked kofta firmer.
Shape the kofta. Divide the mixture into small portions and form each one into a compact oblong patty. Lightly oil your hands if the mixture becomes sticky.
Preheat the skillet. Lightly oil a large skillet and heat it over medium-high heat.
Cook in batches. Arrange the kofta in the hot skillet without overcrowding. Cook for approximately 12–14 minutes, turning every few minutes so the outside browns evenly.
Check for doneness. Continue cooking until the center of the kofta reaches 160°F (71°C) on an instant-read thermometer.
Serve. Plate your Beef Kofta with Greek Yogurt Sauce while the kofta is warm, adding the cool yogurt sauce on the side or drizzling it over the top.