Begin by heating the milk in a small saucepan or microwave. Once this simmers, remove from the heat and allow to cool slightly. You want this to be lukewarm (113°F/45C if using a thermometer). When this is a good temperature, sprinkle the yeast over this along with 1 teaspoon of caster sugar and give a stir. Let this sit for 10 minutes until the yeast is foamy.
Meanwhile, place the flour in the bowl of a stand mixer fitted with a dough hook. Add 2 tablespoons of caster sugar, 2 tablespoons of brown sugar and salt. Stir this and set aside.
Once the yeast is ready, add this to the flour, immediately followed by the eggs and half of the butter (¼ cup) reserving the rest for filling the rolls. Knead this on a medium low setting until it comes together in a soft dough and continue kneading for about 10 minutes until the dough is pulling itself from the sides of the bowl.
Transfer the dough into a lightly buttered mixing bowl and cover with plastic wrap. Set in a warm place to rise for about 90 minutes until doubled in size.
While the dough is proofing, mix the chopped apples with the cinnamon, remaining caster sugar and brown sugar in a small saucepan and cook over a medium heat for about 10 minutes. This helps the apples to release their juices.
After 10 minutes, drain the apples, reserving the cinnamon sugar liquid. Set the apples aside in a bowl to cool and return the cinnamon syrup to a small saucepan. Bring this to a boil for another 10 minutes to help this thicken and set it aside to cool.
Once the dough has doubled in size, turn this out onto a floured work surface and roll out into a roughly 9" x 15" rectangle.
Dollop the remaining ¼" softened butter over the top of the dough and spread to cover as evenly as possible. Then, sprinkle the apple pieces and pour the cinnamon syrup. If the syrup has become too thick and sticky, simply reheat this to loosen it.
Once the fillings have been added, roll the dough up from the short (9") side. Slice the ends off by about an inch to remove any dough without filling and then divide into 9 slices.
Line a 9" square baking pan or other similar sized tin with parchment paper and place the rolls inside. Cover with a clean tea towel and leave in a warm place for an hour to proof for a second time.
Meanwhile, preheat the oven to 350°F (175C/155C Fan) and make your frosting. Simply whisk together the powdered icing sugar, vanilla and milk until you have a thick frosting with a drizzle-able consistency.
Once the rolls have risen for a second time, bake these in the preheated oven for 20 to 25 minutes, until lightly golden.
Remove from the oven and allow to cool for about 10 minutes, before frosting.