Pour oil to a depth of 3 to 4 inches and preheat to 370 degrees F.
In a mixing bowl, combine the cornmeal, flour, baking powder, baking soda, salt, and cayenne. Whisk together the eggs, green onions, milk, and 1/4 cup of the vegetable oil. Add the egg mixture to the flour mixture, stirring just until combined.
Carefully spoon 6 or 7 rounded tablespoonfuls of the batter into the preheated oil. Fry the hushpuppies, in batches, until golden brown, stirring constantly for overall browning, 2 to 3 minutes per batch.
Remove the hushpuppies and drain on a paper towel-lined plate. Season with additional salt and Essence, if desired. Serve immediately.
Yield: about 2 dozen