Warm the milk over medium-high heat, swirling once or twice, until steaming with small bubbles at the sides, a skin on top, and light rippling, about 3 minutes. Reduce heat; continue for another minute without boiling. Remove from heat, measure ⅓ cup into the stand mixer bowl.

Stir the remaining milk with ¼ cup flour in the saucepan, whisking until smooth. Cook over medium while whisking, until thickened to a mashed potato consistency, about 1 minute. Transfer to the stand mixer bowl.

Whisk in cold water, then eggs, ensuring combined. Check bowl temperature—refrigerate to room temperature if still warm. Add yeast and remaining 4 cups flour; mix on low for a coarse dough, about 5 minutes. Let rest for 5 minutes. Add sugar and salt; mix on medium-low until smooth, elastic, pulling from sides, 15 to 20 minutes.

On low speed, add butter piece by piece, letting each absorb before adding more, until dough is supple, 10 to 15 minutes.

Butter a large bowl, transfer dough, seam side down. Cover, let rise until 50% larger, 1 to 1½ hours. The refrigerate at least 8 hours, up to 24 hours.

To make the cinnamon filling, process cookies, sugar, cinnamon, and salt until fine crumbs form in a food processor. Mix with butter until smooth paste forms. Set aside.

Butter the bottom and sides of a 9x13-inch pan, line with parchment, butter parchment paper as well. You'll also want to butter some foil for use as a cover when you put them in the oven.

On floured surface, roll chilled dough to ⅜-inch thick yielding a rectangle about 20x12 inches.

Spread filling across the dough, leaving half-inch border. Roll snugly from the long edge, and resting it seam side down. If dough is soft, chill 10-15 minutes.

Use dental floss to cut and discard ends of your rolled up dough then cut the remaining roll into 15 pieces, making sure each piece is about the same thickness. Arrange in the prepared pan and cover with the buttered foil. Let rise until almost doubled again, 30 to 60 minutes.

Preheat oven to 350°F and prep a center rack. Cover with foil; bake 15 minutes. Remove foil, bake until golden, 15 to 20 minutes more. Cool 15 minutes, loosen sides, remove from pan to cooling rack.

To make the icing, mix cream cheese, sugar, vanilla until smooth. Add butter and whisk until light and thick. Spread over cooled buns.