Add ⅓ cup all-purpose flour to medium shallow bowl and set aside.
Crack 1 large egg directly into large mixing bowl. Whisk egg until well beaten.
Add 1 pound ground chicken, 3 scallions, ¼ cup chopped fresh parsley, 3 tablespoons mayonnaise, 3 tablespoons breadcrumbs, ½ teaspoon salt, ¼ teaspoon freshly ground black pepper, and 1 teaspoon garlic powder to bowl with egg. Use hands to mix all chicken patty ingredients together until well combined.
Portion chicken patty mixture into 6 equally-sized scoops. One by one, quickly roll each scoop between palms of hands to shape into smooth ball. Place each ball in shallow bowl of flour.
After shaping each scoop, roll balls of chicken patty mixture in flour, coating all sides of each ball. One by one, gently press floured ground chicken balls into flattened patty shape and set aside on large plate.
When all chicken patties have been pressed, heat large skillet over medium-high heat. Add 3 tablespoons neutral oil to skillet and swirl or tilt pan as needed to distribute oil across entire surface. Continue heating skillet until oil is hot and shimmery.
Once oil is ready, carefully place prepared chicken patties in skillet, working in batches so as not to overcrowd pan. Cook patties undisturbed 3 to 4 minutes, or until golden brown on underside.
Once browned, flip chicken patties over. Cook patties on remaining side 2 to 4 minutes, or until internal temperature reaches 160° Fahrenheit when checked with internal meat thermometer.
Transfer cooked chicken patties to bed of paper towels to allow excess oil to drain. For crispier chicken patties, place wire cooling rack over bed of paper towels and transfer chicken patties directly from skillet to wire rack.
Repeat cooking process with any remaining uncooked chicken patties. When all chicken patties have been cooked, serve patties warm as desired.