Pour 3 to 4 cups vegetable oil into a large sauté pan or pot to a depth of about 1/2 inch. Heat over medium until the oil reaches 355°F (use a thermometer and monitor as it heats).
In a medium bowl, whisk together the all-purpose flour, 1/4 cup granulated sugar, baking powder, and salt until combined.
In a separate bowl, whisk the buttermilk and melted butter together. If the dough seems too dry, add up to 1 tablespoon additional buttermilk.
Pour the buttermilk mixture into the dry ingredients and stir with a fork just until the dough comes together; do not overmix.
Use a 1/2 tablespoon measure or your hands to roll the dough into small balls (about 35 total).
When the oil reaches 355°F, carefully add about half of the dough balls to the pan without overcrowding. Fry until the bottoms are golden, then flip and fry the other side until golden brown.
Use a spider strainer or tongs to remove the donut holes and drain them on a paper towel–lined baking sheet.
Place 1/3 cup granulated sugar in a shallow dish and roll the warm donut holes in the sugar to coat. Repeat with remaining dough balls.