Make the Apple Butter: Core and cut the apples into chunks. Place in a heavy pot along with the apple cider, apple vinegar, brown sugar, and salt. Cook over medium heat for 30 minutes, stirring frequently, until apples are soft (at this point, you can use an immersion blender to blend the apples directly in the pot if you prefer it super smooth. Or you can just skip this step entirely).
Reduce the heat to low and continue cooking, stirring regularly, until the mixture has reduced significantly and is thick, glossy, and a deep amber colour, about 2 hours. When you drag a spoon through the apple butter, it should leave a trail that holds for a moment before slowly filling in. Let cool completely before using or transferring to an airtight container. You'll have leftover apple butter. Store it in the refrigerator for up to 2 weeks.
Prepare the dough: Bloom the yeast - stir a teaspoon of sugar into the lukewarm milk, followed by the yeast. Leave to proof for 10 minutes, until bubbly.
In a large bowl, (or the bowl of your stand mixer) mix together the flour, salt, and remaining sugar.
Pour the eggs and bloomed yeast mixture into the flour. Mix until a rough dough forms.
If doing by hand, turn the dough onto a lightly floured surface and begin kneading. Knead for about 5 minutes to start developing the gluten - the dough will start to get smoother and less sticky as you work it.
Start adding the room temperature butter to the dough, bit by bit, making sure each bit is combined before adding more. Continue kneading for another 10-20 minutes until the butter is fully incorporated and the dough is elastic. Check for full gluten development using the window pane test.
If using a stand mixer, mix on a slow speed for 4 minutes then increase the speed to medium. Mix for a further 10-12 minutes, adding the butter little by little, making sure each bit is fully incorporated before adding more, until the dough comes away cleanly from the side of the bowl.
Shape the dough into a ball and place into a clean greased bowl. Cover with plastic or a kitchen towel and let it rise for around 60 minutes or until double in size.
This dough is easier to work with when chilled. So you might need to put it in the fridge for about 30 minutes before rolling out so the butter can solidify a little bit. It'll make it easier to roll out.
Make the filling: using a hand mixer, beat the butter and brown sugar together until pale. Stir in the cinnamon and apple butter. Set aside until ready to use.
Roll out the dough: lightly flour a work surface and roll the chilled dough into a rectangle roughly the size of an A3 sheet of paper (12 x 16 inches). Spread half the filling all over the top of the dough.
Perform a letter fold: fold the dough into thirds like a letter - fold the bottom third up to cover the middle third, then fold the top third down to cover the middle third. Transfer to a baking sheet and cover with plastic film, then place into the freezer for 10 minutes to firm up.
Repeat steps 12 and 13: roll out dough into a rectangle once again and spread the rest of the filling. Perform another letter fold. Place in the freezer for another 10 minutes to firm up.
Shape the Buns: Cut the dough widthwise into 2-inch (5cm) strips. Working one strip at a time, make two equal cuts lengthwise, leaving the top intact, so you end up with three equal strands. Braid the strands together, then roll the braid up along its length to create a bun. Repeat with the remaining strips.
Grease a 12 cup muffin tray and place the buns into the cups. Cover and allow to rise for 1 hour, until double in size.
Preheat the oven to 375°F (190°C).
Make the egg wash: whisk the egg and whole milk together until combined. Brush the buns with the egg wash. Transfer to oven and bake for 15-20 minutes, or until buns are golden brown and cooked through.
Meanwhile, make the sugar glaze: In a small saucepan, combine the sugar and water and bring to a boil. Reduce the heat to a simmer and cook, stirring occasionally, until the sugar has dissolved and the syrup has thickened.
Brush the buns with the syrup while still hot. Dust with icing sugar, if desired. The buns should stay soft for up to 3 days when stored in an airtight container.