In a glass measuring cup, warm milk and butter in microwave at 50% power for 2 minutes, stirring halfway. Milk will be warm to the touch but not scalding hot.
Whisk in egg with a fork until combined, then stir in yeast and sugar.
Place currants in a large microwave-safe mixing bowl and microwave 100% for 40 seconds, stirring halfway.
Stir in flour and salt to combine, then stir in milk mixture until a dough is formed.
Cover with a tea towel and place in a warm spot for 1 hour to rise.
Prepare 2 baking trays with parchment paper. Preheat oven to 400F (200C) or 375F(190C) if using fan-forced. Reposition oven racks to lowest and middle positions.
Split the dough into 20 small buns and place them on baking trays. Cover with tea towels and let rise for 20 minutes.
Place in oven and bake 15 minutes until golden brown.
Remove and let cool for 15 minutes.
Mix milk and icing sugar with a fork and drizzle over each bun with a spoon. Allow to dry 20 minutes, then serve.