Preheat the oven to 475 degrees F.
Into a bowl, sift together the self-rising flour, cake flour, baking powder, baking soda, sugar, and salt. Using your fingers or a pastry cutter, work the butter into the flour until the mixture resembles coarse crumbs. Add the buttermilk, cheese, and green onions and, using your hands or a rubber spatula, stir just until the buttermilk and flour come together to form a dough, being careful not to overmix.
Lightly flour a work surface with the all-purpose flour. Turn out the dough onto the surface and press into a disk about 1/2-inch thick and 8 inches in diameter. Using a 3-inch round cutter dusted in flour, cut into rounds. (Be sure to press straight and downward when cutting the dough -- a twisting motion will prevent the dough from rising.)
Reform the scraps in order to make 7 biscuits. Place on a small baking sheet and brush the tops with the melted butter. Bake until golden brown, 10 to 12 minutes.
Remove from the oven and serve hot or at room temperature.