- Whisk flour, yeast, and salt together in bowl of stand mixer. Whisk milk, sugar, egg yolks, and vanilla together in 4-cup liquid measuring cup until sugar has dissolved. Using dough hook on low speed, slowly add milk mixture to flour mixture and mix until cohesive dough starts to form and no dry flour remains, about 2 minutes, scraping down sides of bowl as needed.
- Increase speed to medium-low, add butter, 1 piece at a time, and knead until butter is fully incorporated, about 4 minutes. Continue to knead until dough is smooth and elastic and clears sides of bowl, 10 to 12 minutes.
- Transfer dough to lightly floured counter and knead by hand to form smooth, round ball, about 30 seconds. Place dough seam side down in lightly greased large bowl or container, cover tightly with plastic wrap, and let rise until increased in size by about half, 11/2 to 2 hours. Place in refrigerator until dough is firm, at least 1 hour or up to 24 hours. (If dough is chilled longer than 1 hour, let rest at room temperature for 15 minutes before rolling out in step 5.)
- Combine brown sugar, flour, butter, egg white, cinnamon, and salt in bowl. Measure out and reserve 1 tablespoon filling.
- Grease 8½ by 4½-inch loaf pan. Press down on dough to deflate, then transfer to lightly floured counter. Press and roll dough into 18 by 14-inch rectangle, with long side parallel to counter edge. Spread all but 1 tablespoon reserved filling over dough, leaving ½-inch border around edges.
- Roll dough away from you into firm cylinder, keeping roll taut by tucking it under itself as you go. Pinch seam closed, then reshape cylinder as needed to be 18-inches in length with uniform thickness. Position cylinder seam side up and spread reserved filling over top. Fold cylinder on top of itself and pinch ends to seal.
- Gently twist double cylinder twice to form double figure eight. Place loaf seam side down in prepared pan, pressing dough gently into corners. Cover loosely with greased plastic and let rise until loaf is level with lip of pan, 1½ to 2 hours.
- Adjust oven rack to lower-middle position and heat oven to 350 degrees. Gently brush loaf with egg wash and bake until deep golden brown and loaf registers 190 to 195 degrees, 40 to 45 minutes, rotating pan halfway through baking. Let loaf cool in pan for 15 minutes. Remove loaf from pan and let cool completely on wire rack, about 3 hours, before serving.
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