Make the dough: Have liquid ingredients at 80°F and all other ingredients at room temperature. Place ingredients in pan in the order recommended by manufacturer. Select dough/manual cycle. Do not use delay timer. Check dough consistency after 5 minutes of kneading. The dough should be in a soft, tacky ball. If it is dry and stiff, add water, 1/2 to 1 tablespoon at a time. If it is too wet and sticky, add 1 tablespoon of flour at a time. After dough cycle is complete, continue with next step.
Turn dough onto lightly floured surface; punch down to remove air bubbles. Divide dough into 2 equal parts. On lightly floured surface, roll each half into a 12x6-inch rectangle. Cut out doughnuts with 2½-inch doughnut cutter. Place on lightly floured or greased cookie sheets. Cover; let rise until indentation remains after touching.
In large heavy pot, heat 4 inches of vegetable oil to 400°F (use candy thermometer to check temperature). Fry doughnuts, a few at a time, turning once, until golden brown. Drain on absorbent paper towels. Glaze doughnuts (see below) or shake warm in a paper sack with sugar.
Make Glaze: In small bowl, combine all ingredients and blend until smooth. Dip warm doughnuts into glaze. Place on racks until glaze hardens.
Doughnuts are best eaten the same day.