In a small bowl, combine 1 tablespoon of the brown sugar with 1/4 cup warm water (around 110°F) and sprinkle the active dry yeast over top. Let it sit for 5 minutes until foamy—this proves the yeast is alive and active. While the yeast activates, sift together the flour, cinnamon, nutmeg, allspice, and salt in a separate bowl. I like to sift the spices with the flour rather than adding them later, as it distributes them more evenly throughout the dough and prevents clumpy spice pockets.
In a large mixer bowl, combine the milk with the remaining 5 tablespoons brown sugar and stir until the sugar dissolves. Add the room-temperature eggs and melted butter to the milk mixture, stirring well to combine. Pour in the foamy yeast mixture from Step 1, and stir to combine all wet ingredients. This creates a smooth, enriched base that will develop into a tender, slightly sweet dough.
Add the sifted flour mixture from Step 1 to the wet ingredients in the mixer bowl, stirring with the dough hook on low speed until a shaggy dough forms. Once the flour is incorporated, increase speed to medium and knead for 8 to 10 minutes until the dough becomes smooth and elastic. The dough should be slightly sticky but hold together; if it's too wet, add a tablespoon of flour at a time. Fold in the 1/3 cup mixed fruits by hand after kneading, distributing them evenly throughout.
Form the kneaded dough into a ball and place it in a lightly oiled bowl, turning it to coat all sides. Cover with a damp towel or plastic wrap and let rise in a warm place for 1 hour, or until roughly doubled in size. I find that placing the bowl near a sunny window or in a turned-off oven with the light on provides consistent, gentle warmth for rising.
Preheat your oven to 350°F. Punch down the risen dough to release excess gas, then divide it into 12 equal pieces and shape each into a tight ball, placing them on a parchment-lined baking sheet. Divide any remaining small pieces of dough into thin strips to create crosses for the top of each bun. Press one strip horizontally and one vertically across the top of each bun, sealing the ends with slight pressure.
Cover the shaped buns loosely with plastic wrap or a damp towel and let them rise for 30 to 45 minutes in a warm place until puffy and nearly doubled. Just before baking, brush the tops and sides of each bun with the egg white mixed with a splash of water—this creates a light, glossy finish. Bake at 350°F for 20 to 30 minutes, until the buns are golden brown and sound hollow when tapped on the bottom.
While the buns bake, combine 1/4 cup sugar, 1/2 cup water, and 1 1/2 teaspoons vanilla essence in a small saucepan. Bring to a boil and simmer for 3 to 5 minutes until slightly thickened and syrupy. As soon as the buns come out of the oven, brush them generously with the hot glaze, which will seep into the warm buns and add a subtle sweetness. Let the buns cool for 10 minutes before serving—this allows the glaze to set and makes them easier to handle.