Add 1 cup flour, yeast, sugar, and salt to stand mixer bowl; whisk to combine, set aside.
Combine milk, water and oil in a microwave-safe dish or sauce pan; heat to 120-130°F. Add warm milk mixture to flour mixture; mix with paddle attachment for 4 minutes on medium speed. Add egg; beat 1 minute.
Switch to dough hook attachment. Gradually add enough remaining flour to make a soft dough, and knead until smooth and elastic, about 5 to 7 minutes. (NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.) Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about an hour.
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 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.