Preheat oven to 400°F (200°C). Line a rimmed baking sheet with parchment paper and lightly brush with olive oil.
In a large bowl, combine breadcrumbs and milk; let sit for 5 minutes until absorbed. Add ground chicken, egg, feta, minced garlic, oregano, dill, salt, pepper, and 1 tablespoon olive oil. Mix gently with your hands until just combined.
Roll mixture into 1 1/2-inch meatballs (about 20) and place on the prepared baking sheet. Drizzle with remaining 1 tablespoon olive oil.
Bake for 18–20 minutes, until meatballs are lightly golden and cooked through (internal temperature reaches 165°F).
While meatballs bake, make tzatziki: In a medium bowl, stir together Greek yogurt, grated cucumber, lemon juice, grated garlic, dill, and salt. Refrigerate until ready to serve.
Serve meatballs warm with tzatziki on the side for dipping, or spoon tzatziki over the top. Garnish with extra dill if desired.