Beat together butter and granulated and brown sugars with an electric mixer on medium-low speed until creamy, about 3 minutes. Beat in whole eggs 1 at a time, then salt and 1 teaspoon cinnamon, until combined. Beat in milk and vanilla. Add flour a little at a time, beating until incorporated. Divide dough into 4 balls. Roll 1 ball into a 12-inch log; wrap in parchment and refrigerate, along with remaining 3 balls of dough, until firm, about 1 hour.
Combine cocoa powder and remaining 1 teaspoon cinnamon and sprinkle over work surface. Roll out 1 ball of dough into a 6-by-12-inch rectangle in cocoa mixture. Flip rectangle over onto lightly floured work surface so that 1 long side is facing you; brush off excess cocoa mixture. Place log of dough, also with a long side facing you, in center of rectangle and wrap rectangle around log to enclose completely. Firmly squeeze and massage wrapped log of dough to adhere rectangle to log. Repeat process with second ball of dough, rolling it out into an 8-by-13-inch rectangle, then with final ball, rolling it out into a 10-by-14-inch rectangle. Refrigerate log until firm, about 1 hour, 15 minutes. Log can be stored in refrigerator up to 1 day.
Preheat oven to 350 degrees. Lightly beat egg white. Brush outside of log with egg white and roll in turbinado sugar, patting to adhere. Refrigerate until sugar is set, about 10 minutes. Slice log into 1/4-inch-thick rounds and transfer to parchment-lined baking sheets. Bake until slices are golden around edges (rings may separate slightly), 18 to 20 minutes. Let cool completely. Slices can be made 3 days ahead and stored at room temperature.