Combine the water and milk in a medium saucepan and warm over low heat until about 100 degrees F (but no more than 110 degrees). Remove from heat and sprinkle the yeast and a pinch of sugar and flour over the surface of the liquid. Set aside without stirring, until foamy and rising up the sides of the pan, about 30 minutes.
Whisk the butter, egg yolk and vanilla into the yeast mixture.
Whisk the flour, the remaining sugar, salt, nutmeg, cinnamon and ginger in a large bowl. Make a well in the center of the flour and stir in the yeast mixture with a wooden spoon to make a thick, shaggy, and slightly sticky dough. Stir in currants. Turn the dough onto a lightly floured work surface and knead until soft and elastic, about 8 minutes. Shape into a ball.
Brush the inside of a large bowl with butter. Put dough in bowl, turning to coat lightly with butter. Cover with plastic wrap. Let rise at room temperature until doubled in size, about 1 hour 30 minutes. (If you have a marker, trace a circle the size of the dough on the plastic, and note the time to help you keep track.)
To form the rolls: Butter a 9 by 14-inch baking pan. Turn the dough out of the bowl and pat into a rectangle about 16 by 8 inches. Divide the dough into 12 equal portions, about 2 ounces each, with a pizza wheel or bench scraper. (If you don't have a scale, divide the dough in half lengthwise, then in half crosswise. Cut each of those four sections into 3 equal-sized rolls.)
Tuck the edges of the dough under to make round rolls and place them seam-side down in the prepared pan, leaving a little space in between each roll. Cover the pan with buttered plastic wrap and set aside in a warm place until the rolls rise almost to the rim of the pan and have more than doubled in size, about 45 minutes.
Meanwhile, position a rack in the center of the oven and preheat to 375 degrees F.
Remove the plastic wrap and brush the tops of the buns with beaten egg. Bake rolls until golden brown and puffy, and an instant read thermometer inserted into the center of the rolls registers 190 degrees F, about 25 minutes.
For the glaze: Stir together confectioners' sugar, milk, lemon zest and vanilla until smooth. Transfer icing to a zip bag or pastry bag, and make a small cut in the corner of the bag. Ice buns in a thick cross shape over the top of the warm buns.