In a large mixing bowl, combine the sourdough discard, warm milk, sugar, melted butter, egg, and vanilla extract. Mix well until combined.
In another bowl, whisk together the flour, instant yeast, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until a dough forms.
Knead the dough on a lightly floured surface for about 5-7 minutes, until smooth and elastic. Alternatively, you can use a stand mixer with a dough hook for about 5 minutes on medium speed.
Place the dough in a greased bowl, cover it with plastic wrap or a kitchen towel, and let it rise in a warm place for 1 hour, or until doubled in size.
Meanwhile, prepare the filling by mixing together the brown sugar and cinnamon in a small bowl.
Preheat your oven to 350°F (175°C).
Once the dough has risen, punch it down and roll it out on a floured surface into a rectangle (about 16x12 inches).
Spread the softened butter over the rolled-out dough, then sprinkle the cinnamon-sugar mixture evenly on top.
Starting from the long edge, tightly roll the dough into a log. Cut the log into 12 equal pieces using a sharp knife or dental floss.
Place the cinnamon rolls in a greased 9x13-inch baking dish, spacing them slightly apart. Cover and let them rise for another 30 minutes.
Bake in the preheated oven for 25-30 minutes, or until golden brown.
While rolls are baking, prepare the icing by whisking together powdered sugar, milk, and vanilla extract until smooth.
Once out of the oven, allow the rolls to cool for a few minutes before drizzling with icing.