In a large mixing bowl, combine the warm milk, granulated sugar, and active dry yeast. Stir gently and let it sit for 5-10 minutes until a foamy layer appears on the surface, indicating the yeast is active.

Add the all-purpose flour and salt to the yeast mixture. Mix with a wooden spoon or use a stand mixer with a dough hook attachment until a shaggy dough forms. Incorporate the softened unsalted butter and continue mixing or kneading for 8-10 minutes until the dough is smooth, elastic, and pulls away cleanly from the sides of the bowl.

Lightly grease a clean bowl with oil. Place the dough in the bowl, turning it once to coat all sides. Cover the bowl tightly with a clean kitchen towel or plastic wrap. Let the dough rise in a warm, draft-free place for 1 to 1.5 hours, or until it has visibly doubled in size.

Gently punch down the risen dough to release the air. Turn it out onto a lightly floured work surface. Flatten the dough into a rectangular shape. Evenly sprinkle the shredded Gruyère cheese, finely diced cooked ham, and minced fresh chives (if using) over the surface of the dough. Fold the dough over itself and gently knead for 1-2 minutes to ensure the filling is well distributed, being careful not to overwork the dough.

Shape the dough into a uniform log and place it seam-side down into a greased 9x5 inch loaf pan.

Loosely cover the loaf pan with a clean kitchen towel or plastic wrap. Let it rise in a warm place for another 30-45 minutes, or until the dough has nearly doubled in size and looks puffy.

About 15 minutes before the end of the second rise, preheat your oven to 375°F (190°C). In a small bowl, whisk together the large egg yolk and tablespoon of milk to create the egg wash.

Gently brush the top of the risen loaf with the prepared egg wash. Bake for 35-40 minutes, or until the crust is deeply golden brown and the bread sounds hollow when tapped on the bottom. If the top begins to brown too quickly, loosely tent the loaf with aluminum foil for the remaining baking time.

Remove the bread from the oven and let it cool in the loaf pan for 10 minutes. Then, carefully transfer the loaf to a wire rack to cool completely before slicing and serving. This ensures proper texture and prevents gumminess.