Heat a medium saucepan over low heat until it reaches at least 100 degrees F and exceeds no more than 110 degrees F.
Add the water and the milk to the saucepan, mix well, and heat until it is warm.
Transfer the milk mixture away from the heat and lightly sprinkle it with the dry yeast, the extra sugar, and the extra flour.
Set the yeast mixture aside, without stirring, until it is foamy and rising up the sides of the pan, about 30 minutes.
Whisk the butter, the egg yolk, and 1 1/2 teaspoons of the vanilla into the yeast mixture.
In a large bowl, whisk the remaining all-purpose flour, the remaining sugar, the fine salt, the nutmeg, the cinnamon, and the ginger together.
Make a well in the center of the flour mixture and stir in the yeast mixture until the dough is thick, shaggy, and slightly sticky.
Stir the currants into the dough and mix well, but don't overwork the dough.
Lightly flour a flat and clean surface.
Transfer the dough to the worksurface and knead until it is soft and elastic, about 8 minutes.
Shape the dough into a ball.
Brush the inside of a bowl with the extra butter.
Transfer the dough ball to the buttered bowl and turn to lightly coat it with the butter.
Cover the bowl with plastic wrap and let the dough rise at room temperature until it has doubled in size, about 1 hour 30 minutes.
Butter a 9x14-inch baking dish.
Transfer the dough to a flat and clean surface and use your hands to pat it into an 8x16-inch rectangle.
Use a pizza wheel or a bench scraper to divide the dough into 12 equal portions that weigh about 2 ounces each. If you don't have a scale, divide the dough in half lengthwise, then in half crosswise, and cut the four sections into three equal-size rolls.
Tuck the edges of the dough under to make round rolls and transfer them, seam-side down, to the prepared pan, leaving a little space between each roll.
Cover the rolls with buttered plastic wrap and let them sit in a warm place until the rolls rise almost to the rim of the pan, about 45 minutes.
Adjust a rack in the center of the oven.
Preheat the oven to 350 degrees F.
In a small bowl, add the egg and beat well.
Discard the plastic wrap and brush the tops of the buns with the beaten egg.
Bake the buns until they are golden-brown and puffy and an instant-read thermometer inserted into the center of the thickest part registers 190 degrees F, about 25 minutes.
Transfer the buns to a wire rack.
While the rolls are baking, in a medium bowl, add the confectioners' sugar, the milk, the lemon zest, and the remaining vanilla and mix until it is smooth.
Transfer the glaze to a pastry bag, move it to the top of the bag, and make a small cut in the empty bottom corner of the bag.
Use the pastry bag to glaze the buns in a cross-shaped fashion.
Serve.