Cream sugar and shortening. Beat in cold potatoes, eggs, cooled milk, yeast, water, salt and lemon extract. Mix in flour adding just enough to make a soft dough. Knead well.
Place dough in greased bowl; cover and let rise in warm place until doubled in bulk.
Roll out dough to 1/2 inch thickness. Cut with a well-floured cutter. Let rise uncovered about 30 minutes. Deep fry in oil heated to 375°F. Drain on paper towel.
To make glaze, gradually add enough boiling water to powdered sugar and vanilla to make thin glaze. Drizzle over doughnuts.