Step 1Make the buns: Put the currants, raisins, candied orange peel, rum and 2 tablespoons water in a small bowl. Stir well, then let stand until the liquid completely soaks into the fruit, 2 to 8 hours.
Step 2Half an hour before making the buns, refrigerate the flour and sugar in separate bowls and let the butter stand at room temperature.
Step 3In the bowl of a stand mixer fitted with the dough hook, mix the whole eggs and 3 tablespoons milk on the lowest speed just to blend. Add the flour, salt, yeast, cardamom, cloves, nutmeg, cinnamon and allspice and mix on the lowest speed until incorporated into the liquid. Scrape the bowl, then raise the speed to high and mix for 2 minutes. With the machine running, rain in the sugar slowly, letting it incorporate before adding more; this should take about 5 to 7 minutes. The dough will start to come off the hook at this point and get a little softer.
Step 4Scrape the bowl and add all the butter. Mix on high to fully incorporate the butter, scraping the bowl once or twice, for about 10 minutes. The dough will form a mass around the hook and pull away from the sides of the bowl with a slight thwacking sound. The dough is done when it is glossy, smooth and moist but not sticky.
Step 5Add the dried fruit mixture and mix on low speed until evenly distributed, 1 to 2 minutes. Coat a large bowl or container with cooking spray. Place the dough in the bowl and roll it around to coat with the oil. Cover with plastic wrap and let rise in a warm place (75 to 85 degrees) for 30 minutes.
Step 6Turn the dough out onto a lightly floured work surface. Gently flatten the dough and fold into thirds as you would a letter. Turn it 90 degrees and fold it the same way again. Return the dough to the bowl seam-side down, cover, and let rise until nearly doubled in size, about 30 minutes. Transfer to the refrigerator and chill for 12 hours.
Step 7Turn the dough out onto a lightly floured work surface. Cut the dough into about 20 pieces (50 grams each). Form each piece into a ball by folding in the edges of a cut dough piece toward the center and pull and pinch the seams shut. Put the ball seam side down on the work surface, cup it with one hand and roll until smooth. Place the balls 2 inches apart on a large baking sheet lined with parchment paper. Loosely cover with plastic wrap and let rise in a warm place until doubled in size, about 1 1/2 hours. The gifs below show how to form tight balls of dough that won't unravel when baked in the oven.
Step 8Heat the oven to 350 degrees. Whisk the egg white with the remaining 1 tablespoon milk in a small bowl to make an egg wash. Using a pastry brush, gently brush each risen bun with the wash.
Step 9Bake until dark golden brown, about 15 minutes. Cool completely in the pans on wire racks, about 30 minutes.
Step 1Make the icing: In a small bowl, stir together the powdered sugar, milk and orange blossom water to create a thick consistency that should not run when piped.
Step 2Fill a pastry bag fitted with a quarter-inch plain round tip, or a resealable plastic bag with a quarter-inch hole snipped in one corner, with the icing. Pipe a line along each row of buns, then repeat in the other direction to create crosses. Let stand until the icing is set. Serve immediately.