Mix water, yeast, flour, and salt into a rough dough. Cover and rest for 30 minutes.
Perform a set of stretch-and-folds, going around the bowl at least once. Rest 20 minutes, then repeat once more.
Cover and refrigerate for 12–24 hours.
Make the filling: Mix together the granulated sugar, brown sugar, and cinnamon.
Slice 4.5 tbsp of the butter into small rectangles and keep chilled. Reserve the remaining 1.5 tbsp for later.
Prepare the pan: Grease an 8×8-inch pan with butter or oil.
Shape the dough: Transfer the dough to the pan and gently stretch it to fit.
Arrange the sliced butter evenly over the dough, then sprinkle with about ⅔ of the cinnamon sugar.
Fold the dough like a letter—fold the top down, rotate the pan 180°, and fold again. Flip so the smooth side is facing up.
Final proof: Cover and let rise for 2–4 hours, until puffy.
Preheat oven to 400°F.
Melt the remaining 1.5 tbsp butter and drizzle over the dough. Sprinkle with the remaining cinnamon sugar, then gently dimple the surface with your fingertips.
Bake at 400°F for 20–25 minutes, until deeply golden. If browning too quickly, tent with foil.
Make the glaze: Using a hand mixer or stand mixer, beat cream cheese and butter until smooth. Add powdered sugar and vanilla, mixing until fully combined.
If needed, add milk 1 tbsp at a time until you reach your desired consistency.
