Activate the Yeast: In the bowl of a stand mixer, combine the warm milk, yeast, and 1 tablespoon of the granulated sugar. Let it sit for 5-10 minutes, until it becomes foamy.

Mix the Dough: Add the remaining granulated sugar, flour, salt, and eggs to the yeast mixture. Using the dough hook attachment, mix on low speed until a shaggy dough forms, about 2-3 minutes.

Add the Butter: With the mixer still running on low, add the softened butter cubes one at a time. Once all the butter is incorporated, increase the speed to medium and knead for 10-15 minutes, until the dough is smooth, elastic, and pulls away from the sides of the bowl.

First Rise: Transfer the dough to a lightly greased bowl, cover with plastic wrap, and refrigerate for at least 8 hours or overnight. This long, cold fermentation is crucial for developing the rich flavor and texture of the brioche.

Prepare the Filling & Assemble: On a lightly floured surface, punch down the chilled dough and roll it into a large rectangle, about 12x16 inches. In a small bowl, mix the brown sugar and cinnamon. Brush the melted butter over the entire surface of the dough, then sprinkle the cinnamon-sugar mixture evenly over the top.

Shape the Loaf: Tightly roll the dough into a log, starting from the long side. Transfer the log, seam-side down, into a lightly greased 9x5-inch loaf pan.

Second Rise: Cover the pan with a towel and let the loaf rise in a warm place for about 1-1.5 hours, or until it has almost doubled in size.

Bake the Bread: Preheat your oven to 350°F (175°C). Whisk together the egg and milk for the egg wash and brush it over the top of the risen bread. Bake for 40-45 minutes, or until the top is golden brown and a thermometer inserted into the center reads 200°F (93°C).

Serve: Let the cinnamon brioche bread cool in the pan for 10 minutes before transferring it to a wire rack to cool completely. Slice and serve warm.