In a bowl of a stand mixer, combine milk and yeast; let stand until yeast softens, 5 to 10 minutes. Add whole egg, granulated sugar, butter, salt, nutmeg, cinnamon, and orange zest. Beat on medium speed with dough hook until blended.
Blend in 2 3/4 cups flour. Beat on medium speed until dough is smooth and stretchy, 10 to 12 minutes. Add just enough flour (about 1/4 cup) so dough is only slightly tacky. Add orange peel and currants, pick up dough, and mix with your hands to distribute fruit. Return dough to bowl, cover, and let rise in a warm place until doubled, about 1 1/4 hours.
Punch down dough. With floured hands, shape into 16 smooth rounds. Evenly space rounds in 2 buttered 8- or 9-in. square pans.
Cover loosely and let rise in a warm place until doubled and puffy, about 40 minutes. Meanwhile, preheat oven to 400°. Brush buns with beaten egg. Bake until deep golden, 13 to 15 minutes. Let cool in pans at least 30 minutes.
In a small bowl, stir together juices and powdered sugar until smooth. Spoon into a small, heavy-gauge plastic bag, snip a hole in a corner, and squeeze icing onto buns to form large Xs.
*Buy from gourmet food stores and amazon.com, get the recipe on sunset.com, or substitute orange marmalade: Heat 18 oz. thick-cut marmalade until bubbling, strain well, and substitute 1/4 cup of the syrup for the granulated sugar in step 1 and 1/2 cup orange pieces for the candied orange in step 2; add 1/4 cup extra flour in step
Note: Nutritional analysis is per bun.