For the dough, measure all of the ingredients except the currants and peel into the bowl of a mixer fitted with the hook attachment or in a large bowl. Mix the dough on low speed until it comes together, then increase the speed by one and knead until the dough is smooth and elastic, about 5 minutes (the dough is soft and should stick to the bottom of the bowl). If mixing by hand, stir the ingredients with a wooden spoon until the dough comes together, then turn out onto a lightly floured work surface and knead until smooth. Towards the end of kneading, add the currants and mixed peel and knead in. Scrape the dough into a lightly oiled bowl, cover with plastic wrap and let rise for 75 to 90 minutes, until doubled in size.
Turn the risen dough out onto a lightly floured work surface and divide into 12 even pieces. Shape each piece into a ball by rolling between your hands while it remains on the work surface. Place the rolled buns in a greased 9-x-13-inch pan, leaving space between them. Cover the pan with plastic wrap and let the buns rise for 45 minutes.
Preheat the oven to 350F. Uncover the buns and bake for 25 to 30 minutes, until a rich brown on top. While still hot from the oven, prepare the glaze.
For the glaze, bring the sugar, water and vanilla up to a simmer, stirring until the sugar is fully dissolved. Brush this syrup over the still-hot buns, until it has all been used. Let the buns cool completely in the tin.
For the icing, stir the icing sugar and together until a thick consistency suitable for piping (add a few more drops of milk, if needed). Pour this into a small piping bag and pipe crosses on top of each bun, letting the icing set for an hour before serving.
The buns will, keep, well wrapped, for a full day.