Warm 2/3 cup milk to 105–115°F (warm to the touch, not hot). Pour the warm milk into a small bowl, sprinkle the package of dry active yeast over the surface, stir in the 1/3 cup sugar, and let sit 5–10 minutes until foamy. If the mixture does not become foamy, the yeast may be inactive—discard and start with fresh yeast.
While the yeast proofs, melt the 1/3 cup butter and let it cool until warm (not hot).
In the bowl of a stand mixer, whisk together the 1/4 teaspoon salt and 2/3 cup of the flour (reserve the rest of the flour for later).
Add the foamy yeast-milk mixture, the warm melted butter, and 1 whole egg to the mixer bowl. Mix on low speed just until the wet ingredients are combined with the dry.
Fit the mixer with a dough hook. With the mixer running on low, add the remaining flour gradually—about 2 to 2 1/3 cups total—adding 1/4 cup at a time until a soft, slightly sticky dough forms. Use the lower amount of flour first and add more only if the dough is too sticky to handle.
Knead the dough in the mixer on medium-low speed with the dough hook for 5–7 minutes, until the dough is smooth, elastic, and pulls away from the sides of the bowl but remains slightly tacky.
Transfer the dough to a lightly greased bowl, turn once to coat the top, cover with plastic wrap or a clean kitchen towel, and let rise in a warm, draft-free place until doubled in size, about 1 to 1 1/2 hours.
Punch the risen dough down, turn it out onto a lightly floured surface, and roll it into a rectangle suitable for braiding (a roughly 12 x 8–10 inch rectangle works well).
To form a braid: trim the short ends so they are straight, then cut 1-inch-wide strips along both long sides of the rectangle, leaving a center panel about 3–4 inches wide. The center panel will remain flat while you braid the side strips over it.
Fold the strips over the center panel alternately from each side to form a braid, overlapping each strip about halfway across the center. Tuck the ends under to finish and transfer the braided dough to a parchment-lined baking sheet. Cover loosely and let rise 20–30 minutes until puffy.
Preheat the oven to 375°F. Meanwhile, make the egg wash by whisking together the reserved 1 egg yolk and the 1 teaspoon water.
Brush the braided dough with the egg-yolk wash, then bake in the preheated oven 18–22 minutes, or until the braid is golden brown and cooked through. Remove from the oven and cool on a wire rack before serving.