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Dissolve the yeast in 125ml (1/2 cup) water. If using fresh yeast, go straight on to step If using dry yeast, leave it in a warm place for 10-15 minutes until it's foamy/bubbly.
Place the flour either in a bowl or directly on your work surface and create a little well in the middle. Pour the yeast mixture in, and, using your fingers, mix it all up to form a rough dough.
Knead the dough for a good 5 minutes until it's all stretchy and smooth. Place in a lightly greased bowl and cover with a cling film or tea towel and leave in a warm place to rise for 1 1/2 hours until practically double in size.
In the meantime, get the butter out and place it on a large sheet of baking paper. Fold the baking paper over in half, completely covering the butter (see video). With a rolling pin, beat down on the butter to flatten. This is actually much easier than you think. Roll it slightly to help it to flatten into a rectangle roughly about 23cm (9") x 15cm (6"), give or take. An inch either way doesn't really matter. Place the rectangular disc of butter back in the fridge until needed.
Grease the cake pan and line with baking paper. Grease the baking paper.
Preheat the oven to 180˚ (350˚F/16˚C Fan).
When the dough has had its 1 1/2 hours, tip it onto a lightly floured surface. Flatten the dough, sprinkle some flour over it and start rolling it out into a rectangle measuring about 28cm (11") x 20cm (8"). Essentially, we want rolled out dough that's about 5cm (2") wider than the butter, all around.
Now get the butter out of the fridge, and still in the baking paper, place it on the dough to see if the dough is big enough with 5cm (2") extra all around. If not, roll out the roll a little bit more.
Working quickly, especially if you have warm hands (mine are always cold!), take the wrapping paper off the butter and place the butter centrally on the dough.
Sprinkle most of the sugar on the butter slab, leaving about 2 Tbsp aside.
Now fold the edges of the dough over the butter, overlapping the dough. You will have to stretch the dough to make it overlap. That will be easy to do, as it's a stretchy dough.
Roll the dough out to form a rectangle. Then fold the top third of the dough down, and fold the bottom third over the top you've just folded down. Take a look at the video, and this will be clear.
Then you need to fold it in the same way in the other direction too (see the video). If it's not long enough to fold, just pat it down hard or roll EVER SO SLIGHTLY to make it a little longer. Now, you've done 1 fold.
Repeat this step (roll into a rectangle, fold, fold) 2 more times. You can do it a 4th time if you fancy.
After the 3rd fold, roll out the dough gently, to form a round shape as much as possible. Don't worry too much about it being a perfect round, you can fit it into the cake pan.
Fit the dough into the cake pan.
Brush the milk all over the top of the unbaked kouign amann.
Using a bread lame or a scoring tool (blade), score some shallow cuts all over the kouign amann.
Sprinkle half the remaining sugar all over the kouign amann dab the extra butter all over, and bake in the oven for 30 minutes.
When done, sprinkle with a little more sugar, let it cool slightly, cut into slices and serve.