Add milk to a small sauce pan and heat over medium-high heat until warmed to 90°F to 100°F, and remove from heat.
Add active dry yeast and sugar, stirring to combine. Allow to bloom 5 to 10-minutes until active and frothy.
While yeast mixture rests, add the fine sea salt and bread flour in the bowl of a stand mixer and whisk to combine.
Add the yeast mixture and beaten egg to the dry ingredients and knead on low speed until combined, approximately 5 minutes. Scrape down the sides of the bowl as needed.
Once you have a rough dough, increase the speed to medium-low and add the softened unsalted butter one tablespoon at a time until thoroughly combined, about 5 to 7 minutes.
Turn the dough onto an unfloured work surface and shape into a smooth ball.
Place dough into a large mixing bowl, cover and rest for 1-hour until doubled in size. Prepare the Caramelized Onion and Gruyere Compound butter while the dough is rising.
Caramelized Onion and Gruyere Compound Butter
While the dough is rising, peel and halve the onions from pole to pole (top to bottom). Use a mandolin slicer set to ⅛ to ¼-inch thick and slice the onions into half rounds.
Heat a large skillet over medium-high heat and melt 4 tablespoons of butter. Once melted, add the sliced onions and several pinches of kosher salt.
Reduce the heat to medium and cook for 30 to 45 minutes until browned and caramelized. Be sure to stir every several minutes to make sure the onions don't burn and they caramelize evenly.
Remove the onions from the heat and allow to cool. Add the Worcestershire sauce and taste for seasoning.
In a medium bowl, combine the grated gruyere, caramelized onions, 1 stick of butter, and a pinch of kosher salt until evenly incorporated. (If the onions are too warm still and the butter melts, cover and refrigerate or freeze for several minutes until the butter sets back up.)
Bread Assembly and Second Proof
Towards the end of the first rise, lightly grease a 4 ½ x 9-inch loaf pan and set aside.
Once the dough has risen, turn it out onto a floured work surface and punch it down. Divide the dough into 12 equal pieces.
Working one piece of dough at a time, roll it into a ball. Use a rolling pin to flatten to ¼-inch thick, and about 4 to 5-inches wide.
Place 1 tablespoon of the prepared compound butter on the dough round, and fold the dough in half.
Add the folded dough to your greased loaf pan with the seam side facing up. Add another tablespoon of the prepared compound butter onto the exposed edge of the dough. (It's ok if some of the compound butter squeezes out the top when folding).
Repeat with the remaining dough and compound butter until the loaf pan is full.
Cover and allow to rise for an additional hour.
Baking
Towards the end of the second rise, preheat the oven to 350°F.
After the dough has risen the second time, whisk together the egg white and water in a small bowl and gently brush over the top of the loaf. Sprinkle flaky sea salt over the top.
Bake for 30 to 35 minutes, or until golden brown and the internal temperature reaches at least 195°F on an instant read thermometer.
Remove and allow to cool for about 10 minutes in the pan.
Garnish
Transfer the bread to a wire cooling rack to cool completely, and sprinkle with thinly sliced chives and more flaky salt if needed.
Notes
You can substitute equal amounts of All-Purpose Flour for the Bread Flour. All purpose flour has slightly lower amount of protein in it, so the gluten network won't be quite as strong as if you use bread flour, but you will still have really good results.StoringIf you make this recipe a day or more ahead of time, cover it and keep it in the refrigerator to ensure it stays as fresh as possible.ReheatingTo reheat the whole loaf, do the following:
Remove the bread from the refrigerator and allow it to warm up on the counter for an hour or more.
Wrap the loaf in tin foil so the top doesn't burn. (You don't want to cook this any more; you're just looking to warm it through.)
Heat in a 325°F oven for 15 to 20 minutes until totally warmed through.
If you just have some pieces or just part of the loaf leftover (you'll be lucky to have any!) instead of the entire loaf, do the following:
Separate the remaining pieces (if needed)
Place uncovered on a foil-lined baking sheet and reheat at 350°F for 5 to 8 minutes.