Step 1

Make the dough: Coat a large bowl with oil and set aside. Combine the 1 cup milk, yeast, and the 1 teaspoon sugar in a small bowl and let stand until bubbly. Combine the flours, remaining sugar, raisins, candied peels, salt, and spices in the bowl of a standing mixer fitted with a dough hook (or in a large mixing bowl) and mix on low speed. Add the butter, 2 eggs, and the yeast mixture and continue to mix until a sticky dough forms—about 3 minutes. Transfer the dough to a lightly floured surface and knead by hand until smooth—about 5 minutes. (If dough has been combined by hand, increase kneading time to 10 minutes.) Form the dough into a ball, place it in the prepared bowl, and turn to coat all sides with oil. Cover with a clean, damp kitchen towel and let rise in a warm, draft-free place until it doubles in volume—about 1 hour.

Step 2

Shape the buns: Line a baking pan with parchment paper and set aside. Punch the dough down, transfer to a lightly floured surface, and knead for 3 minutes. Divide the dough into 12 equal-sized pieces, about 3 1/2 ounces each. Shape each piece into a ball and place the balls about 1 inch apart in three rows of four on the prepared pan. Cover and let rise until the buns double in volume and touch one another, about 1 1/4 hours.

Step 3

Bake the buns: Preheat oven to 500°F. In a small bowl, combine the egg yolk with 1 tablespoon water. Using a pastry brush, lightly brush the mixture on the top of each bun. Place buns in the lower third of the oven and reduce oven temperature to 400°F. Bake until golden brown, about 20 minutes. Transfer to a wire rack. In a small bowl, combine the confectioners' sugar, remaining milk, and vanilla. Stir until smooth. When buns have cooled slightly, drizzle a horizontal line across each row of buns followed by a vertical line to form a cross on the crown of each bun.