Prepare the brioche cinnamon roll dough (3 hours or 1 day ahead of time):

1. Activate the yeast: Place the milk in a microwave-safe bowl and heat in 15-second intervals until warm to the touch, but not hot or steaming (about 45 seconds to 1 minute total). Transfer the warm milk to the bowl of a stand mixer fitted with the dough hook. Add the yeast and sugar, stirring to combine. Let sit until foamy, about 5-10 minutes.
2. Mix the brioche dough: Add the eggs, flour, and salt to the yeast mixture. Mix on medium speed until a sticky dough forms. With the mixer running, add the butter 2 tablespoons at a time until fully incorporated. Continue kneading on medium speed for 10 minutes, until the dough forms a smooth, tight ball and bounces back when pressed.
3. Rest: Transfer the dough to an oiled bowl and cover with plastic wrap and a kitchen towel. Let the dough rest for 12 hours or overnight in the refrigerator (preferred), or in a warm spot in your kitchen for 2 hours, until doubled in size.

Prepare and bake the chai cinnamon rolls (1 hour before baking):

1. Roll out the brioche dough: Place the dough on a lightly floured surface. If it rose overnight, it may feel stiff but will roll out easily. Using a rolling pin, roll the dough into a large rectangle, about 12×16 inches.
2. Add the brown sugar and chai spices: Leaving a ½-inch border along the short sides, spread the softened butter evenly over the dough. Sprinkle the brown sugar, chai spice blend, and salt evenly over top.
3. Roll and slice the chai spice cinnamon rolls: Starting from a short edge, roll the dough into a tight log with a chai spice swirl running down the middle. Pinch the edge of the dough to seal the log shut. Using a sharp serrated knife (or unflavored dental floss for cleaner cuts), slice the log in half, then slice each half in half again to create 4 sections. Cut each section into 3 pieces to make 12 rolls.
4. Proof the rolls: Transfer the chai cinnamon rolls to a greased 9×13-inch baking pan. Cover with a kitchen towel and let them rise in a warm spot until doubled in size, about 30 minutes to 1 hour.

Bake, frost, and serve:

1. Bake the chai cinnamon rolls: Preheat the oven to 350°F with a rack positioned in the center. Once hot, bake the chai cinnamon rolls for 20-25 minutes, until golden brown and cooked through. Remove from the oven and let cool slightly.
2. Prepare the espresso icing: Meanwhile, as the rolls bake, prepare the icing. In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese and butter on medium-high speed until light and fluffy, about 3-4 minutes. Add the powdered sugar, chai spice, and salt, mixing on low until combined. Gradually add the brewed espresso or coffee, 1 tablespoon at a time, until the icing reaches your desired consistency—I usually like 3 tablespoons.
3. Ice the rolls and serve: When the rolls are cool enough to handle, generously spread the icing over the top. Serve immediately with coffee or hot cocoa. Enjoy!