Make the Dough: Warm the milk by microwaving it for about 60 seconds. Use a kitchen thermometer to check that it's between 105-110℉. If you don't have one, just ensure the milk is warm to the touch, but not scalding hot.
Add the warm milk, active dry yeast, and brown sugar to the bowl of a stand mixer. Whisk, cover with a towel, and place in a warm area for 5-10 minutes, or until the milk mixture is foamy (that's how you know the yeast is activated).
Add the egg, softened butter, salt, sourdough discard, and flour to the bowl of a stand mixer (alternatively, you can use a wooden spoon and a large mixing bowl if you're not using a stand mixer) with the dough hook attachment on. Mix the ingredients on medium speed until the flour is combined, making sure to scrape the sides of the bowl with a rubber spatula to incorporate all the flour.
Knead the dough (either with your hands or with the stand mixer) on medium for 7-9 minutes. The dough is done kneading when you press your finger on the dough and it doesn't stick to your finger.
Form the dough into a ball and return it to the bowl. Cover the bowl with plastic wrap or a kitchen towel, and place in a warm area to rise for 1.5 to 2 hours (or once the dough has doubled in size).
Prepare the Cinnamon Rolls: Preheat oven to 350℉. Dust a clean surface with flour. Roll out the dough into a 12" by 18" rectangle. Spread the butter evenly on top of the dough. Next, sprinkle the brown sugar and cinnamon evenly over the buttered dough.
Roll the dough length-wise into a tight cylinder and cut into 12 equal-sized rolls. For best results, use unflavored dental floss to cut the rolls. You may also use a sharp serrated knife. To ensure the cinnamon rolls are equal in size, cut the roll in half, then each half in half again, then finally, each half in thirds.
Place the rolls in a greased 13x9" baking dish. Bake for 30-35 minutes until the rolls are fully baked and are golden brown on top.
Make the Frosting: While the cinnamon rolls are baking, make the frosting. With a hand or stand mixer, combine the cream cheese, butter, powdered sugar, and vanilla extract until smooth. If you'd like to thin the frosting out, add a tablespoon of milk. Drizzle over the cinnamon rolls while they are still warm. Serve immediately and enjoy!