Preheat the oven to 475°F.

Sift the self-rising flour, cake flour, baking powder, baking soda, sugar, and salt together into a large bowl. Using your fingers or a pastry cutter, work the cold butter into the flour until the pieces are pea size. Add the cheddar and green onions. Pour the heavy cream into the mixture, and with your hands or a rubber spatula, stir just until the cream and flour come together to form a dough.

Sprinkle some of the all-purpose flour on a work surface and place the dough on top of the flour. Using a lightly floured rolling pin or your hands, shape the dough into a 1/2-inch-thick disk about 8 inches in diameter. Using a sharp 3-inch round cutter dusted with flour, cut out 6 dough rounds. Be sure to press straight down when cutting the dough—a twisting motion will prevent the dough from rising.

Place the biscuits on an ungreased baking sheet and brush the tops with the melted butter. Bake until golden brown, 10 to 12 minutes.