In a small bowl, stir together yeast and warm water. Allow to rest for 5-10 minutes, until foamy.
In a large mixing bowl fitted with a dough hook or paddle attachment, combine scalded milk, shortening, sugar and salt. Stir to dissolve the ingredients in the warm milk. Cool the mixture to a lukewarm temperature.
Add the softened yeast mixture to the milk mixture. Add the egg, 1 cup of flour, cinnamon, nutmeg and allspice. Beat on low speed for 30 seconds, then scrape down the sides of the bowl with a spatula. Add remaining flour and the currants (or raisins); beat on medium speed until the dough comes together and pulls away from the sides of the bowl (about 1-2 minutes). If the dough seems too wet, gradually add a spoonful of flour at a time until the dough comes together and pulls away from the sides of the bowl. Continue mixing the dough for 2 more minutes (or knead by hand on the counter for 2 minutes).
Lightly grease a bowl with oil. Transfer the dough to the bowl and turn to coat. Cover tightly with plastic wrap and let the dough rise in a warm place until doubled in size (about 2 hours).
Punch down the dough. Divide the dough into 12 even pieces; shape into smooth balls. Arrange the dough balls in a greased 9 x 13-inch baking dish.
Cover the dish and let the dough rise again until puffy, about 60-90 minutes.
Remove the cover, brush the tops of the buns with egg wash, and bake on the middle rack in a 350° F oven for 18-22 minutes. Let cool in the pan on a wire rack.
Once the buns are cool (or just slightly warm, but not too hot), prepare the glaze. In a small bowl, whisk together the confectioners’ sugar, vanilla and milk. The glaze should be thick, but thin enough so that you can pipe it onto the buns. If it’s too thick, add a small amount of milk to thin. If it’s too thin, add a small amount of confectioners’ sugar until it reaches the desired consistency. Transfer the glaze to a piping bag or a sandwich bag with the corner snipped off; pipe a cross onto each bun.