In a separate bowl, beat the egg with the buttermilk and melted butter, then stir in the chopped onion and green onion.
Mix the wet ingredients into the dry ingredients until just combined. Let rest for 5 minutes for a better texture.
Lower the temperature to 380°F and cook for 4-5 minutes, then carefully flip each hush puppy and cook for an additional 3-5 minutes until golden and crispy.