Savory Ham and Cheese Brioche Buns

Prepare the Brioche Dough

1. In the bowl of a stand mixer, combine flour, yeast, sugar, and salt. Mix briefly.
2. Add the eggs and warm milk. Mix on low speed with the dough hook until a shaggy dough forms, about 2-3 minutes.
3. Increase speed to medium-low and knead for 5 minutes until the dough starts to come together. Gradually add the softened butter, one cube at a time, allowing each addition to be incorporated before adding the next.
4. Continue kneading on medium speed for 10-15 minutes, until the dough is smooth, elastic, and pulls away from the sides of the bowl. It should be soft and slightly sticky but manageable.

First Rise and Chill

1. Transfer the dough to a lightly oiled bowl, cover with plastic wrap, and let it rise in a warm place for 1.5 to 2 hours, or until doubled in size.
2. Once doubled, gently punch down the dough, reform it into a ball, and cover it tightly with plastic wrap. Refrigerate for at least 3 hours, or preferably overnight. Chilling makes the dough much easier to handle.

Assemble the Buns

1. On a lightly floured surface, roll out the chilled brioche dough into a large rectangle (approximately 12x18 inches).
2. Spread the Dijon mustard evenly over the dough, leaving a small border around the edges.
3. Sprinkle the diced ham evenly over the mustard, then follow with the shredded Gruyère cheese.
4. Roll the dough tightly from one long side to the other, forming a log.
5. Using a sharp knife or unflavored dental floss, cut the log into 12 equal slices (about 1.5 inches thick).
6. Arrange the slices in a greased 9x13 inch baking pan or on a baking sheet lined with parchment paper, leaving some space between them.

Second Rise and Bake

1. Cover the pan loosely with plastic wrap and let the buns rise in a warm place for 45-60 minutes, or until puffy and almost doubled in size.
2. Preheat your oven to 375°F (190°C).
3. In a small bowl, whisk together the egg and water for the egg wash. Gently brush the tops of the risen brioche buns with the egg wash.
4. Bake for 18-22 minutes, or until the buns are golden brown and cooked through. If they start to brown too quickly, you can tent them loosely with foil.
5. Remove from the oven and let cool slightly before serving. These are best served warm.