Stir together warm milk, brown sugar, and yeast in bowl of standing mixer and let stand until foamy, about 5 minutes. (If it doesn’t foam, discard and start over.) Add 3 3/4 cups flour and salt and mix with dough hook at low speed until dough is smooth and very soft, about 7 minutes.
Transfer dough to a work surface and knead by hand 2 minutes, adding more flour as necessary, a little at a time, to make a soft, slightly sticky dough. Form dough into a roughly 1 1/2-inch-thick rectangle and chill, wrapped in plastic wrap, until cold, about 1 hour.
Prepare and shape butter:
After dough has chilled, arrange sticks of butter horizontally, their sides touching, on a work surface. Pound butter with a rolling pin to soften slightly (butter should be malleable but still cold). Scrape butter into a block and put on a kitchen towel, then cover with other towel. Pound and roll out on both sides until butter forms a uniform 8- by 5-inch rectangle. Chill, wrapped in towels, while rolling out dough.
Roll out dough:
Unwrap dough and roll out on a lightly floured surface, dusting with flour as necessary and lifting and stretching dough (especially in corners), into a 16- by 10-inch rectangle. Arrange dough with a short side nearest you. Put butter in center of dough so that long sides of butter are parallel to short sides of dough. Fold as you would a letter: bottom third of dough over butter, then top third down over dough. Brush off excess flour with pastry brush.
Roll out dough:
Turn dough so a short side is nearest you, then flatten dough slightly by pressing down horizontally with rolling pin across dough at regular intervals, making uniform impressions. Roll out dough into a 15- by 10-inch rectangle, rolling just to but not over ends.
Brush off any excess flour. Fold in thirds like a letter, as above, stretching corners to square off dough, forming a 10- by 5-inch rectangle. (You have completed the first "fold.") Chill, wrapped in plastic wrap, 1 hour.
Make remaining "folds":
Make 3 more folds in same manner, chilling dough 1 hour after each fold, for a total of 4 folds. (If any butter oozes out while rolling, sprinkle with flour to prevent sticking.) Wrap dough tightly in plastic wrap and chill at least 8 hours but no more than 18 (after 18 hours, dough may not rise sufficiently when baked).