Heat olive oil in a large skillet over medium-high heat. Once hot, add onion and cook until softened, stirring occasionally, about 2 to 3 minutes. Stir in garlic, cooking for another 30 seconds to 1 minute, until fragrant.
Add in Sprouts grass-fed beef. Cook until completely cooked through, about 5 minutes, breaking up bigger chunks with a wooden spoon.
Drain any excess fat. Stir in spices, tomato paste, and water. Reduce heat to medium-low.
Cook for another 5 minutes, stirring occasionally.
While the meat is cooking, stir together ingredients for the sauce. Set aside.
Pile beef up on pita breads, topping with tomato, onion, cucumber, green onion, feta, and sauce. Serve immediately with lemon wedges.