In a large bowl, combine the ground chicken, garlic, mint, parsley, egg, salt, pepper, panko, Greek seasoning, and feta. Mix gently just until combined.
Use a cookie scoop to portion the mixture evenly, then roll each portion into a ball by hand. Wetting your hands with water first prevents sticking.
Heat oil or butter in a large, deep skillet over medium heat. Working in batches, sear the meatballs on each side until browned (they don't need to cook through yet). Remove and set aside.
In the same pan, melt the butter. Add the vermicelli and cook, stirring, until deeply golden brown and fragrant.
Stir in the rice and toast for 1 minute, then pour in the chicken broth.
Nestle the seared meatballs back on top of the rice. Cover and cook over medium-low heat, leaning closer to low, until the liquid is fully absorbed, about 20 minutes. If not all meatballs fit in the pan, bake the extras on a tray until they reach an internal temperature of 165°F.
Remove from heat and let rest, covered, for 5 minutes. Top with fresh parsley before serving.