Preheat the oven to 475 degrees F. 

Coat a large bowl with oil; Set aside. 

Line a large baking sheet with parchment paper for when you are ready to make the buns.

Whisk 1/2 cup milk, yeast, and 1 teaspoon of sugar in a small bowl and allow to stand until frothy.

Meanwhile, combine the flours, remaining sugar, raisins, candied peels, spices, and salt in the bowl of a stand mixer fitted with the dough hook. Give a quick mix on low speed until combined.

Add the butter, 2 eggs, and the yeast mixture to the flour. Mix for about 3 minutes or until a sticky dough forms.

Turn the dough out onto a clean lightly floured surface and knead by hand until smooth, about 5 minutes. Roll into a ball and place in prepared bowl, flipping to coat evenly with oil. Cover with a damp cloth and allow to rise in a warm space until doubled in size, about 1 hour.

Punch down the dough and turn it out onto a floured surface again. Knead for 3 minutes before dividing into 12 equal portions. 

Form into buns and place about 1" apart from one another on prepared baking sheet. Cover again and allow to rise another hour - to hour and half, or until doubled in size.

Combine the remaining egg yolk with 1 tbs of water and brush gently over each bun. 

Place buns in oven and reduce heat to 400 degrees. Bake for 20 minutes, until golden. Allow to cool on a wire rack.

When cool, whisk the remaining milk with the confectioners’ sugar and vanilla extract until smooth. Place in a ziploc bag, snip off the edge and ice with a horizontal line across each bun followed by a vertical line. Allow to set and serve.