Preheat oven to 375°F (190°C). Slice the top off a garlic head, drizzle with olive oil, wrap in foil, and roast for 35 minutes. Cool slightly, squeeze out cloves, mash into a paste. Set aside.
Grate cucumber, squeeze out moisture. Mix with Greek yogurt, roasted garlic paste, lemon juice, dill, salt, and pepper. Stir in olive oil. Chill for 30 minutes.
Combine ground chicken, breadcrumbs, onion, garlic, dill, parsley, lemon zest and juice, egg, oregano, salt, and pepper. Flatten 1½ tbsp of mixture, add a feta cube, wrap and roll into a ball. Repeat to form 16–20.
Heat oil in a skillet over medium. Cook meatballs in batches for 10–12 minutes, turning until golden and cooked through (165°F). Drain on paper towels.
Serve meatballs with roasted garlic tzatziki. Garnish with herbs. Enjoy as appetizers, in pita, or over grains.