Heat 1 quart vegetable oil in a deep fryer or large saucepan to 365 degrees F (185 degrees C).

Preheat the oven to 200 degrees F (95 degrees C).

Whisk buttermilk, eggs, and 1/4 cup vegetable oil together in a bowl.

Whisk cornmeal, flour, sugar, baking soda, and salt together in a separate bowl; fold in buttermilk mixture, green onions, and minced onion until just combined.

Carefully drop batter by tablespoon-sized balls into the hot oil in batches of 6 to 8 balls. Fry until golden brown, flipping to cook evenly, 6 to 10 minutes. Transfer hush puppies with a slotted spoon onto brown paper bags to drain. Repeat with any remaining batter.

Transfer hush puppies to a baking sheet; keep warm in the preheated oven until ready to serve.