In a large mixing bowl, whisk the flour, yeast, sugar and salt; set aside.
In the stand of an electric mixer with a paddle attachment or dough hook, mix together the warm milk, ¼ cup of melted & cooled butter, warm brewed coffee, and the egg.
Gradually add the dry ingredients to the wet ingredients, in ½ cup increments, just until the dry ingredients are fully incorporated and the dough starts to pull away from the mixing bowl.
Lightly flour a flat clean surface and turn out out the dough. Knead until smooth, about 5 minutes. You'll know the dough is ready when it springs back a bit when you lightly press a finger into it.
Place the dough in a bowl, cover with a damp cloth and let it rest for 10 minutes.
Meanwhile, lightly grease a 9" x 13" baking dish with butter; set aside.
In a medium bowl, whisk together the brown sugar, instant coffee, and cinnamon.
On a lightly floured surface, use a rolling pin to roll out the dough into a 9" x 12" rectangle.
With a pastry brush, brush ¼ cup of softened butter over the top of the rolled out dough, covering it entirely.
Immediately sprinkle the sugar cinnamon coffee mixture over the entire surface of the dough, pressing the mixture in lightly before the softened butter sets to help prevent it from spilling out. You may have extra sugar cinnamon coffee mixture; set this aside for later.
Gently roll up dough on the long side, pinching the seam along the edge to seal it.
Using a serrated knife, cut the dough into 12 equal size portions and place cut sides up in the prepared baking dish. Cover loosely with a cloth and let the rolls rise in a warm place until doubled in size, about 60 minutes.
Preheat the oven to 350°F.
When the dough has risen, bake at 350°F for 20-25 minutes, just until the tops are lightly golden.
Meanwhile, make the frosting: in the bowl of a stand mixer with a whipped cream attachment, beat together the softened butter and cream cheese until smooth, about 3-5 minutes, scraping down the sides of the bowl as needed.
Beat in the vanilla and coffee. Then add powdered sugar in ½ cup increments to the bowl while the mixer is on low, until the powdered sugar is fully incorporated. Beat in the instant coffee granules just until incorporated.
Once the rolls are done baking, let them cool for about 10 minutes, then spread the frosting evenly over the tops. If you have any extra sugar cinnamon coffee mixture, sprinkle over the tops of each roll. Serve warm.
Notes
Makes 12-15 cinnamon rollsDough Tips:
Warm the milk in the microwave for 30 seconds.
Allow the butter to cool after being melted, but not so much that it starts to solidify again.
Make sure the coffee is warm - not hot and not cold - before using.
Make sure the egg is at room temperature. This will help it mix seamlessly into the dough.
Frosting Tips:
Make sure the coffee is cooled before using it, otherwise, the frosting may curdle.
Take the cream cheese and butter out of the refrigerator about 20-30 minutes before you will use it so that both are softened enough to mix smoothly.
Start with 2 cups of powdered sugar, then gradually add more in 1/4 cup increments until the frosting is the thickness you desire.
Storage: store in an airtight container in the refrigerator for up to 3 days.