In a large bowl, combine the yeast with 2 teaspoons of the sugar and the warmed milk (at 110°F). Stir well and let this mixture sit for 5 minutes—this activates the yeast and ensures your buns will rise properly. While this sits, in a separate large bowl, whisk together the 5 cups of bread flour, bread improver, mixed spice, and cinnamon to evenly distribute the spices throughout. I always sift my spices with flour to avoid clumps that can create uneven flavor distribution in the buns.
Add the activated yeast mixture, egg, softened butter, and remaining sugar (about 1/2 cup plus 2 teaspoons) to the flour mixture. Stir until a shaggy dough forms, then turn it out onto a floured surface and knead for 8 minutes until the dough becomes smooth, elastic, and slightly tacky. The dough should spring back when poked and feel soft but not sticky—this development creates the tender crumb structure these buns are known for.
Transfer the kneaded dough to an oiled bowl, cover it with a damp kitchen towel or plastic wrap, and place it in a warm spot (around 75-80°F works well) for 1 hour. The dough should roughly double in size. I like to place mine in a turned-off oven with the light on—it creates the perfect warm environment without any risk of overheating.
Once the dough has risen, gently punch it down to release excess gas. Divide it into 12 equal pieces and shape each into a smooth ball by tucking the edges underneath. Arrange the balls in a parchment-lined baking dish (roughly 9x13 inches), spacing them close together so they'll touch when they rise—this helps them stay soft and connected. Cover with a damp towel and let rise in a warm place for another 1 hour until they're puffy and touching each other.
About 15 minutes before the second rise finishes, preheat your oven to 200°C (about 390°F). While the oven heats, prepare the cross paste by mixing the 1/2 cup flour with 1/3 cup water in a small bowl until you have a smooth, pipeable consistency. Transfer this mixture to a piping bag fitted with a small round tip. Once the buns have finished their second rise, pipe a cross on top of each bun in a quick, confident motion.
Place the buns in the preheated 200°C oven and bake for 20 minutes until they're golden brown on top and the crosses are set. The buns should sound slightly hollow when tapped on the bottom when done.
While the buns are still hot from the oven, prepare the glaze by heating the apricot jam with 2 tablespoons of water in the microwave for 30 seconds, then stirring until smooth. Brush this glaze generously over each hot bun—the heat will help the glaze set into a glossy finish. Transfer the buns to a wire cooling rack and let them cool slightly before serving. I prefer Bonne Maman jam because it has fewer seeds and creates a smoother, more elegant glaze on the finished buns.