Whisk the yeast into 2 tablespoons warm tap water in a medium bowl, then leave it for about 5 minutes. Look for a cloudy mixture with a little puffiness before continuing.

Stir in ⅓ cup room-temperature water, followed by the flour, cooled melted unsalted butter, 2 tablespoons sugar, and salt; use a spatula or wooden spoon, bring the rough mixture together with several hand kneads in the bowl, then knead on a clean counter for 6 to 8 minutes. The finished dough should feel soft, smooth, and supple; add a light sprinkle of flour only when it sticks to your hands or counter.

Round the dough, lightly flour its surface, return it to the bowl, and cover with a damp kitchen towel in a warm spot for 1 to 1¼ hours. Use its starting size as your reference and wait until the ball has nearly doubled.

During the rise, pile the cold salted butter pieces on parchment, fold the paper over them, and tap with a rolling pin into a roughly ¼-inch-thick layer; open the packet, use a spatula to form a 6-inch square, rewrap snugly without air pockets, and roll in both directions with the parchment folds underneath before refrigerating. Aim for straight edges, filled corners, and uniform thickness, with butter that yields rather than turning sticky.

Gently deflate the risen dough with your fist, move it to parchment, press and stretch it into a 7-inch square, then wrap and roll across the packet in both directions before placing it in the freezer for about 5 minutes; meanwhile, take the butter block out of the refrigerator to soften slightly. The dough should become very firm without freezing, while the butter remains cool and workable.

Unwrap the dough onto a lightly floured counter and save its parchment; expose the butter’s top, invert it onto the dough with its corners pointing toward the middle of each dough edge, discard the butter paper, then bring the dough corners inward and pinch every seam closed. The dough should overlap slightly at the center and completely seal the butter inside.

Lightly flour the dough above and below, tap it with the rolling pin, then roll forward and backward into a rectangle about ¼ inch thick and three times as long as it is wide. Keep the corners square and use additional flour underneath only where sticking occurs.

Sweep away loose flour, scatter a little of the separate ½ cup sugar over the rectangle, and fold the lower third into the middle before bringing the upper third over it. This three-layer letter fold completes the first turn and stacks the enclosed butter and dough.

Give the folded dough a quarter-turn counterclockwise, then repeat the rolling, light sugar scattering, and letter fold, adding flour only if needed. You should now have a compact folded packet with two completed turns.

Enclose the packet in the saved parchment and refrigerate for 45 minutes to 1 hour. Continue when the dough feels firm and relaxed.

Use a little of the 3 tablespoons cooled melted salted butter to brush a 9-inch cake pan, fit fresh parchment across its bottom and up its sides, then distribute the remaining butter over the lining while reserving 1 tablespoon for the top; scatter 2 tablespoons from the separate ½ cup sugar across the bottom and shake the pan gently. Smooth the parchment creases and check that butter covers the entire base and sides, with sugar evenly spread below.

Unwrap the chilled dough, keep the parchment, and set the packet on a lightly floured counter with its flap uppermost and the flap’s free edge on your right. The packet is oriented correctly when lifting that flap would open it toward the left.

Roll another long rectangle, using flour underneath only as necessary, scatter a little more of the remaining sugar over it, fold into thirds, and rewrap for a 20- to 25-minute refrigerator rest. The dough should again feel firm and relaxed before the final rollout.

Roll the unwrapped dough on a barely floured counter into a thin 15-inch square, slide it onto the saved parchment, fold its corners to the center, press the new corners inward, and roll into a 9-inch round; use the paper to invert it into the prepared pan, then coat the top with the reserved tablespoon melted butter and any remaining sugar. The shaped round should fit the pan, with the folded side underneath.

Drape a damp towel loosely over the pan and let the dough rise at room temperature for 45 minutes to 1 hour, or refrigerate the covered shaped pan for up to 12 hours and bake directly from the refrigerator without an additional room-temperature rise. For the room-temperature option, look for a puffed appearance and a modest increase in height.

Position a rack in the middle of the oven and heat to 400°F. Let the oven reach its setting before the pan goes in.

Remove the towel, place the pan in the oven, immediately lower the setting to 350°F, and bake for 35 to 45 minutes; rest the baked cake in its pan for 5 minutes, cover with an inverted wire rack, turn both over, and pull gently on the parchment ends to release the cake before peeling away the paper. Look for deep golden pastry and bubbling caramel at the edges before removing it from the oven, then let the unmolded cake cool completely before cutting wedges.