In a large bowl, dissolve the yeast in warm milk with 1 teaspoon of sugar. Let it sit for 15 minutes until bubbly.
2 ½ teaspoons dried yeast, ¾ cup whole milk, ¼ cup sugar
In a stand mixer bowl, sift together the flour, ground cinnamon, freshly grated nutmeg, and ground mixed spice. Add the yeast mixture, remaining sugar, 1 egg, and yolk. Stir to form a rough dough.
2 ¼ cups all purpose flour, ½ teaspoon ground cinnamon, ½ teaspoon freshly grated nutmeg, 2 large eggs, 1 teaspoon ground mixed spice
Gradually incorporate the butter until fully mixed and the dough becomes sticky. Knead for 5 minutes until smooth and shiny. Add the salt and mix in the raisins and cranberries. Knead for 3 more minutes.
¼ cup soft unsalted butter, ½ teaspoon salt, ¼ cup raisins, ¼ cup dried unsweetened cranberries
Transfer the dough to a lightly greased bowl, cover it, and let it rise for 1 ½ – 2 hours until nearly doubled in size.
Turn the dough onto a floured surface and divide it into 12 equal pieces. Shape each into a smooth bun by rolling on the surface.
Place the buns on a parchment-lined baking tray, leaving a 2 cm gap between them. Cover and let prove for 1 hour until almost doubled in size.
Preheat the oven to 375°F (190°C). Brush the buns with beaten egg, let dry for 10 minutes, and then brush again.
For the cross topping, mix the plain flour, icing sugar, olive oil, and milk in a small bowl until smooth. Fill a piping bag with the mixture and pipe crosses onto the buns. Bake for 20–22 minutes until golden.
½ cup plain flour, ¾ cup icing sugar, 2 tablespoons olive oil, 3 tablespoons milk
For the syrup, combine sugar, water, star anise, whole clove, and vanilla pod (split lengthwise) in a small pan over low heat. Stir until the sugar dissolves, then simmer for 2 minutes until thickened. Remove from heat and discard the spices.
¼ cup sugar, ⅛ cup water, 1 whole star anise, 1 whole clove, ½ vanilla pod
Brush the warm buns with syrup. Let it cool slightly before serving.