Heat 2 inches of oil in a large, deep cast-iron skillet or Dutch oven to 365 degrees F.
Whisk together the cornmeal, flour, sugar, baking powder, baking soda and a good pinch of salt in a medium bowl. Beat in the buttermilk and eggs until combined, then stir in the melted butter and scallions to make a thick batter.
In two batches, drop the batter by the tablespoonful into the hot oil. Fry, turning once, until crisp and golden, about 3 minutes per batch. Drain on paper towels; season immediately with salt.