Blend all the dry ingredients except the currants and fruit peel into a large bowl.
Add eggs, melted butter and warm milk.
Mix the dough with an electric mixer with a dough hook attachment on low speed until the dough comes together.
Add the currants and fruit peel – mix well until fruit is blended and the dough is smooth 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 large wooden spoon until the dough comes together,. If the dough is sticky, drizzle lightly with rice flour, then turn onto a lightly floured work surface using rice flour and knead until smooth.
Shape dough into an evenly level square, about 9 by 9-inches. Cut into equal portions. Shape into rolls and place into a well-greased baking dish (oil fingers lightly to help shape dough).
Cover the pan with plastic wrap and let the buns rise for 45 minutes in a warm area.
Preheat the oven to 375ºF. Uncover the buns and bake for 22 – 25 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 milk 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 before serving. Serve while warm.