Preheat the oven to 400°F. Spray a wire rack with nonstick cooking spray and set over a rimmed baking sheet.
Combine the bread and milk in a large bowl and let sit until the milk is absorbed, about 1 minute. Add the beaten egg, red onion, garlic, cumin, oregano, salt, and a quarter of the ground beef, and mix together until completely combined. Add the rest of the ground beef and mix just until blended.
Divide the meatball mixture into 12 equal portions. Flatten each portion into a circle and place a chunk of feta in the center. Fold the edges of the meat up and around the feta, and roll to form into a round ball.
Place the meatballs on the prepared wire rack set and bake for 15-20 minutes, or until the meatballs are browned and cooked through.
For the tzatziki sauce:
Meanwhile, make the tzatziki. Place the grated cucumber in a kitchen towel and wring out as much moisture as possible. Place the cucumber in a medium bowl. Add the yogurt, minced garlic, lemon juice, olive oil, mint, and salt and stir to combine.
Serve the baked meatballs with the tzatziki sauce, sprinkled with additional finely chopped mint.
Notes
Tip: Make a double batch of these Greek meatballs to take to work or pack in school lunches!