Pour the milk into a small saucepan and place over low heat. Heat the milk until just lukewarm. Take the pan off of the stove and sprinkle the yeast over the surface of the milk. Let stand for 10 minutes or until foamy.
Pour the milk mixture into a stand mixer fitted with a dough hook. Add the eggs, sugar, salt, and 1 1/2 cups of the flour. Mix the dough on medium-low speed until a sticky dough starts to form.
Add the remaining flour in 1/4 cup increments until the dough is smooth, silky, and slightly tacky to the touch. You may or may not have to use all the flour listed to achieve this consistency or you may have to use slightly more.
Take 1/2 cup of the softened butter and cut it into 1-inch cubes. Add the butter to the dough 3-4 cubes at a time, only adding more once the previous addition is fully integrated. This should take a little time, so be patient.
Once the butter is fully integrated, take the dough out of the mixing bowl and place it on a lightly floured surface. Knead the dough by hand for 5 minutes before forming into a ball. Transfer the ball to a large bowl greased with vegetable oil. Cover with a tea towel and let rise in a warm place for 1 1/2 hours or until doubled in size.
Punch the dough down and divide it into 10 evenly sized pieces. Roll each piece out to a long rope, roughly the length of a ruler. Roll the ends of each rope out to form a point. Fold the rope in half and coil the ends around each other 3 or 4 times. Pinch the ends together to seal. Transfer the formed twists to a baking sheet lined with parchment paper and repeat with the remaining dough.
Once all the twists are formed, cover the baking sheet lightly with plastic wrap and transfer to the fridge. Let the twists rise overnight or for 8 hours.
The next morning, preheat the oven to 375°F. Take the brioche out of the fridge and leave them to rise a little further on the kitchen counter while the oven preheats.
In a small bowl, whisk to combine 1 egg and 1 tablespoon of water to make an egg wash. Think of this as makeup for the brioche, if you don't have an extra egg, don't fret it will still taste the same. Using a pastry brush, brush the surface of the brioche with the egg wash.
Transfer the baking sheet to the oven and bake for 10 minutes before rotating the pan. Bake for another 10 minutes or until golden brown. Take the brioche out of the oven and transfer a cooling rack.
Once the brioche is cool enough to handle. Place the remaining butter in a large bowl and place it in the microwave. Melt the butter in 30-second increments until completely melted. In another bowl, whisk to combine the remaining sugar and cinnamon.
Dunk each brioche in the butter and rotate to ensure each is completely covered in butter. Transfer the brioche to the bowl with the cinnamon sugar and roll to ensure complete coverage. Transfer the brioche to a platter and serve immediately with Apple Cider Caramel.