Slice each stick of butter lengthwise into three pieces (6 in all). Sprinkle a little flour over a piece of parchment paper measuring approximately 26 x 15 inches. Place the butter in a horizontal row in the middle of the left half of the parchment paper. Fold the other half of the parchment paper over the butter. This will keep the butter from sticking to the rolling pin. Roll the butter into a flat sheet about 1/8-inch thick. Place the butter sheet into the refrigerator.
In a stand mixing bowl fitted with a dough hook, add flour, 3 tablespoons of sugar, and the salt, and mix on low until combined.
In a small bowl, combine the water, yeast, and 1 teaspoon of sugar, and let it sit for about 15 minutes.
In a medium bowl, add the milk, egg, and vanilla and mix well.
Pour the milk mixture and the yeast mixture into the flour and knead the dough for 7 to 8 minutes.
Turn the dough onto a lightly floured surface and roll dough to an approximately 20 x 20-inch square. Remove the butter from the refrigerator and place 1/2 of the butter sheet over the right side of the square. You may have to remove the butter with a scraper. Cover the butter with the right side of the dough. Place the remaining butter on the bottom half of the dough and fold over with the top half of the dough. Pinch all the sides of the dough to keep the butter from coming out of the sides when you roll the dough. The key is to get thin layers of butter throughout to create a flaky dough.
Roll the dough to about 1/2-inch thick, then fold into thirds just as you would fold a letter. Repeat three more times. The last time, leave it folded like a letter and put the dough in a ziptop bag and refrigerate for at least one hour. I like to put mine in overnight and have dough ready for the next morning.
Remove dough from the refrigerator and place on a floured surface. Roll dough to 1/8-inch thickness, creating a 24 x 24-inch square. Cut the dough into 1-inch strips. With one strip of dough, beginning with the ends, twist the dough in opposite directions. When the dough is spiraled, roll one end continuously to reach the other end, until you have formed a pretty cylinder. Place each danish onto a piece of parchment paper on top of a cookie sheet and allow to rise for one hour.
Remove dough from the refrigerator and place on a floured surface. Roll dough to 1/8-inch thickness, creating a 24 x 24-inch square. Fold the sides, not letting the filling spill out, and seal the pastry together with an egg wash. Place each danish onto a piece of parchment paper on top of a cookie sheet and allow to rise for one hour.
Bake pastries at 375 degrees in a pre-heated oven for 10 to 15 minutes. Remove to a cooling rack and store in an airtight container.
Baker's Note: The scraps that you cut away from the pastry dough to form a square can be used for amazing dough in monkey bread, or you can form them into balls, sprinkle them with cinnamon and sugar and bake until they're golden.