Heat 1 quart vegetable oil in a deep-fryer or large saucepan to 365 degrees F (185 degrees C).
Preheat oven to 200 degrees F (95 degrees C).
Whisk buttermilk, 1/4 cup vegetable oil, and eggs in a bowl.
Combine cornmeal, flour, sugar, baking soda, and salt in a separate bowl. Fold buttermilk mixture, onion, and green onions into cornmeal mixture until just mixed.
Drop 6 to 8 tablespoon-sized balls of batter into the hot oil; fry until each hush puppy is golden brown, turning the hush puppies to cook evenly, 6 to 10 minutes.
Remove hush puppies with a slotted spoon and place on brown paper bags to drain. Repeat with any remaining batter.
Transfer hush puppies to a baking sheet and keep warm in the preheated oven until ready to serve.
