In a large bowl, dissolve yeast in warm water. Add the milk, butter, sugars, nutmeg, salt, eggs and orange zest and 2 cups flour. Beat on medium speed for 3 minutes or until smooth. Stir in enough remaining flour to form a firm dough.
Turn onto a floured surface; knead until smooth and elastic, about 6-8 minutes. Place in a greased bowl, turning once to grease top. Cover and let rise in a warm place until doubled, about 45 minutes.
Punch dough down. Divide in half. On a floured surface, roll out one portion into a 13-1/2x6-in. rectangle. With a sharp knife, cut dough into thirty-six 1-1/2-in. squares.
Using a sharp knife and beginning at the bottom of the lower left square, make two 1-in. vertical cuts 1/2 in. apart. Braid the three attached strips; pinch ends to seal. Place on a greased baking sheet. Repeat with remaining squares and the remaining dough. Cover and let rise until almost doubled, about 40 minutes.
In an electric skillet or deep-fat fryer, heat oil to 375°. Fry doughnuts, a few at a time, for 1 minute on each side or until golden brown. Remove with a slotted spoon to paper towels.
In a large saucepan, combine glaze ingredients. Bring to a rolling boil over medium heat. Remove from the heat. Dip warm doughnuts in glaze. Place on wire racks with waxed paper underneath.
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