In a large bowl, combine the ground chicken, dried oregano, panko, beaten egg, milk, minced garlic, finely chopped red onion, finely chopped parsley, 1/2 teaspoon salt, and 1/4 teaspoon black pepper; mix gently with your hands until evenly combined.
Shape the mixture into 18 small meatballs, about 1 inch each.
Heat the olive oil in a large skillet over medium heat.
Add the meatballs in a single layer (cook in batches if needed) and brown on all sides, turning gently, about 8–10 minutes total, until cooked through.
While the meatballs cook, stir together the Greek yogurt, chopped dill, lemon zest and juice, 1/4 teaspoon salt, ground cumin, and cayenne in a small bowl to make the sauce; taste and adjust seasoning if desired.
Transfer cooked meatballs to a platter, scatter with crumbled feta and cucumber pieces, and sprinkle with chopped parsley or dill.
Serve the yogurt-dill sauce on the side or spoon it over the meatballs.