In a large bowl, dissolve the yeast in warm milk with 1 teaspoon sugar. Let it sit 15 minutes until bubbly.
2 ½ teaspoons dried yeast, ¾ cup whole milk, ¼ cup sugar
In a stand mixer, sift together flour and spices. Add yeast mixture, remaining sugar, 1 egg, and 1 yolk. Mix until a rough dough forms. Gradually add soft butter until fully combined, and the dough is sticky.
2 ¼ cups plain flour, ½ teaspoon ground cinnamon, ½ teaspoon freshly grated nutmeg, 1 teaspoon ground mixed spice, 2 large eggs, ¼ cup soft unsalted butter
Knead 5 minutes until smooth and shiny. Add salt, knead again, then fold in dried fruit. Continue kneading 3 minutes.
½ teaspoon salt, ¼ cup raisins, ¼ cup dried unsweetened cranberries
Place dough in a greased bowl, cover, and let rise 1 ½–2 hours until nearly doubled.
Turn the dough onto a floured surface. Divide into 12 pieces (~50 g each). Roll each into a smooth bun.
Arrange buns on a parchment-lined tray that fits in your air fryer, 2 cm apart. Cover and prove 1 hour until almost doubled.
Brush buns lightly with remaining egg mixture; let dry 10 minutes.
Mix the cross ingredients until smooth. Pipe crosses on buns using a piping bag.
½ cup plain flour, 2 tablespoons olive oil, 3 tablespoons milk, ¾ cup icing sugar
Air fry at 375°F (190°C) for 15 minutes, until golden.
Combine sugar, water, star anise, clove, and vanilla in a small pan. Dissolve sugar over low heat, then simmer 2 minutes. Remove from heat and discard spices.
¼ cup sugar, ⅛ cup water, 1 whole star anise, 1 whole clove, ½ vanilla pod
Brush warm buns with hot syrup. Let cool slightly before serving.