In a small saucepan, melt together the milk and butter. As soon as they melt together, turn off the heat and set that aside.
Add the milk mixture and yeast to the bowl of a stand mixer. Let it sit for 5 minutes.
Add the sugar, eggs, flour, salt, and sourdough discard into the bowl of the stand mixer. Turn the mixer on low speed and mix for 5 minutes.
Cover the bowl with plastic wrap and let it rest at room temperature for 1-2 hours until it has doubled in size.
Once the dough has doubled in size, turn the dough out onto a lightly floured surface.
Roll the dough into a 12 inch by 18 inch rectangle. Spread the softened butter over the dough, leaving a one inch border around the edge of the dough.
Mix together the brown sugar and cinnamon in a small bowl. Sprinkle the mixture over the dough.
Starting with the edge closest to you, roll the dough forward until a log forms. Butter a 9x13 inch pan well.
Cut the log into 12 equal sized pieces and place them in the prepared baking dish. Cover the baking dish with plastic wrap and let it rest for another 1 hour. At this point, you can place it in the refrigerator until you are ready to bake them.
Preheat the oven to 375 degrees F. Remove the plastic wrap and place the baking dish in the preheated oven. Bake for 30-35 minutes or until golden brown on top.
While the cinnamon rolls bake, mix together the butter, cream cheese, powdered sugar, and milk until smooth. For a thinner consistency, add more milk.
Remove the cinnamon buns from the oven and pour the icing over the top. Serve immediately.