Preheat oven to 425°F. Coat a large rimmed baking sheet with cooking spray.
Mix lamb (or beef), egg, 2 tablespoons parsley, mint, 1 teaspoon garlic and salt in a medium bowl. Form into 12 balls and place on the prepared pan. Bake until browned and cooked through, about 20 minutes.
Meanwhile, bring a large saucepan of water to a boil. Cook orzo according to package directions. Drain and let cool for 5 minutes.
Mix oil, dill, lemon zest and lemon juice along with the remaining 2 tablespoons parsley and 2 teaspoons garlic in a medium bowl. Transfer 1 tablespoon of the herb mixture to a small bowl and stir in yogurt and 1/4 cup feta.
Add the orzo to the medium bowl and stir in the remaining 3/4 cup feta. Serve the meatballs with the orzo, yogurt sauce and cucumbers.