Preheat oven to 425°.
Cook the potato in boiling water 15 minutes or until very tender. Drain well; mash potato, and cool.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking powder, baking soda, and salt in a bowl; cut in butter with a pastry blender or 2 knives until the mixture resembles coarse meal. Add mashed potato, cheese, buttermilk, and green onions; stir just until moist.
Turn dough out onto a floured surface, and knead lightly 5 times. Roll dough to a 3/4-inch thickness, and cut with a 3-inch biscuit cutter. Place on a baking sheet. Combine water and egg white, and brush over biscuits. Bake at 425° for 20 minutes or until golden.