Cook potato in boiling water to cover 15 to 20 minutes or until tender. Drain, reserving 1/2 cup cooking liquid. Mash potato well, reserving 1/4 cup.
Dissolve yeast in warm water, stirring well; let stand 5 minutes or until bubbly.
Combine sugar, shortening, and salt in a large mixing bowl; pour scalded milk and reserved potato liquid over mixture, stirring until shortening is melted. Cool mixture to lukewarm (105° to 115°).
Add 1/2 cup flour and yeast mixture; stir well. Add 1 1/4 cups flour, stirring until smooth. Stir in mashed potato, egg, and enough remaining flour to make a soft dough.
Turn dough out onto a floured surface; cover and let rest 10 minutes. Knead dough 5 minutes or until smooth and elastic. Place in a greased bowl, turning to grease top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk. Punch dough down; cover and repeat rising procedure 45 minutes.
Punch dough down; turn out onto a lightly floured surface. Cover and let rest 10 minutes. Shape dough into 3/4-inch balls; divide balls in half, and roil each half into a 3-inch rope.
Drop shaped doughnut sticks, a few at a time, into deep hot oil (365°). Fry until golden brown, turning once. Drain well on paper towels. Serve hot with butter and maple syrup.