Place the 2 sticks of butter in the freezer. (If you would rather not grate the butter, you can make 2 butter slabs instead. Working with one stick at a time, cut into thirds (or quarters) lengthwise, lay the slices next to each other, cover with parchment paper, then roll out into a rectangle (about 8x12 maybe?). Repeat with the 2nd stick. Place in the refrigerator until time to use.)
In a large bowl, stir together the warm water, 1 Tbsp sugar, and yeast.
Add 2 cups flour, 1 Tbsp melted butter, and 1 tsp salt to the yeast mixture. Stir until a very wet, sticky dough forms.
Dust the counter with the remaining 1/2 cup flour and turn the dough onto the floured surface. Knead, folding in the excess flour as needed, until the dough is soft, slightly sticky, and elastic.
Place in a greased bowl, cover, and allow to rise until doubled - about 1 1/2 hours.
Forming the Dough: You will end up folding into thirds 4 times.
Make the sugar blend by combining the sugar and most of the sea salt in a bowl. Taste and add the remaining salt if desired.
Grease a muffin tin then sprinkle some of the sugar blend into each cup. Shake to thoroughly coat then turn the tin over and shake out the excess.
On the floured surface, press the risen dough into a rough rectangle then roll out until 1/8-1/4" thick. Grate 1 stick of butter over the dough, leaving a 1-inch border. Pat down and flatten the butter.Fold the rectangle widthwise into thirds (1st time). Press gently to square out the edges then place on a baking sheet. Cover with Saran wrap and refrigerate for 30 minutes.
Roll the dough into a large rectangle again. Grate the 2nd stick of butter over the surface. Pat down and flatten the butter. Fold the dough into thirds (2nd time), roll back out into a rectangle, then fold in thirds again (3rd time). Roll out again and fold into thirds (4th time). Cover with Saran wrap and refrigerate for at least 1 hour.
Preheat the over to 375°.
Sprinkle a generous amount of the sugar blend onto the work surface. Place the dough onto the sugar and sprinkle more of the sugar blend on top. Roll into a large rectangle about 1/8-1/4" thick, turning and topping with more sugar between rolls.
Trim the edges to make a clean rectangle. Cut the dough into 12 equal squares. Sprinkle more sugar on top.
Lightly pinch each pastry into a crown shape by bringing all 4 corners to the center. Place into the muffin cups.
Sprinkle more of the sugar blend on top. Let rest for 10 minutes.
Bake in the preheated oven until brown and puffed - 25-20 minutes.
Transfer to a cooling rack while still hot. *Make sure you remove while they're still hot or they will become glued to the muffin tin.
Allow to cool for at least 15 minutes before serving.
Notes:
The All-recipe recipe has a good video to help walk through the kouign amann process.