For dough: In the bowl of a stand mixer, whisk together 1¼ cups (300 grams) cold water, yeast, and sugar by hand until yeast and sugar dissolve. Add flour, butter, and salt; using the paddle attachment, beat at low speed until a shaggy dough forms.
Switch to the dough hook attachment. Beat at low speed until dough becomes smooth and forms a ball around dough hook. Increase mixer speed to mediumlow, and beat until dough becomes smooth and elastic, 4 to 6 minutes. Cover with plastic wrap, and let rest at room temperature (67°F/20°C to 72°F/22°C) until dough puffs significantly, 45 minutes to 1 hour.
Using a permanent marker or a pencil, draw an 8-inch square on a large piece of parchment paper; turn parchment over.
For butter block: In a medium bowl, fold and stir butter until it is smooth. Spread butter into an even thickness within drawn square. Fold parchment over butter to cover. Refrigerate until firm but pliable, about 30 minutes, or for up to 2 days.
Punch down dough, and turn out onto a clean surface. Flatten into an 8-inch square, and wrap in plastic wrap. Freeze for 20 minutes.
Remove butter block from refrigerator; it should be cold but flexible. If it’s too soft, return to refrigerator to chill longer. If it’s too hard, let stand at room temperature for a few minutes.
On a lightly floured surface, roll dough into a 16×8-inch rectangle, with one long side closest to you. Unwrap chilled butter block, and place in center of dough. Fold dough over butter block, fully enclosing it and pinching seam to seal. (The seam should be vertical.) Rotate dough 90 degrees so seam is horizontal. Roll dough into a ¼-inch-thick rectangle (about 22×10 inches), with one long side closest to you. (Make sure to maintain the rectangular shape, squaring the dough off with a bench scraper as needed.) Fold dough in thirds like a letter, using a pastry brush to dust off excess flour between each layer. Rotate dough 90 degrees, and repeat rolling and folding procedure.
Wrap dough in plastic wrap, and refrigerate for 30 to 45 minutes. (The dough should relax, and the butter should chill. At this stage, the dough can be refrigerated up to overnight.)
Lightly spray 2 (6-cup) jumbo muffin pans with baking spray with flour; dust with just enough granulated sugar to coat bottom and sides of cups.
On a clean work surface, sprinkle 1⅔ cups (333 grams) sugar in an even layer. Turn out dough onto sugar. With seam facing you, roll dough into a 22×10-inch rectangle. Sprinkle remaining ⅓ cup (67 grams) granulated sugar onto dough in an even layer; fold dough in thirds like a letter.
Rotate dough 90 degrees; roll dough into a 17×13-inch rectangle, dusting dough and surface with sugar as needed. (Work as quickly as you can without tearing or overworking the dough, as the sugar will liquify if left in contact with the dough for too long.) Trim dough to create a 16×12-inch rectangle; cut into 12 (4-inch) squares.
Dust 1 dough square with a little more sugar. Fold corners of square in to center, firmly pressing to adhere. Place your thumbs and forefingers on new corners that have formed, and pinch inward toward center but do not press all the way to center. Place pastry into a prepared muffin cup. Repeat with remaining dough squares. Cover and let rise at room temperature (67°F/20°C to 72°F/22°C) until puffed, 45 minutes to 1 hour.
Preheat oven to 375°F (190°C).
Bake until deep golden brown and caramelized and an instant-read thermometer inserted in center registers 200°F (93°C) to 205°F (96°C), about 25 minutes, rotating pans halfway through baking.
Meanwhile, line 2 wire racks with parchment paper.
Remove pans from oven, and immediately invert pastries onto prepared racks. Let cool completely, flat side up. (This allows the sugar on the underside to harden, preventing the pastry from sinking or collapsing.) Best served same day as made.
