Combine the yeast, warm milk, vanilla extract and sugar in the bowl of a stand mixer fitted with the hook attachment. Allow to sit for 2-5 minutes for the yeast to dissolve. Add in 2 cups of the flour and mix to combine. Now add in the salt and egg. Mix well.
Begin gradually adding in the remaining flour, adding one cup at a time, until the dough begins to pull away from the sides of the bowl. Then, slowly mix in the softened butter, one piece at a time.
Continue to knead the dough until it is well-developed and can pass the windowpane test. This will take 8-10 minutes. Once done, cover the dough and let it proof until it has doubled in size.
While the dough is proofing make the cinnamon sugar butter. Mix together the softened butter, brown sugar and cinnamon. Set aside.
Once the dough is ready to be shaped, line a 9x13 baking dish with parchment paper and preheat the oven to 350°F.
On a lightly floured surface, roll out the dough into a 10×15-inch rectangle. Evenly spread the cinnamon sugar butter over the dough. Begin with the long end and roll the dough into a log. Slice into 12 equal-sized rolls and arrange them in the prepared dish. Cover and allow to proof for 30 minutes, or until they become puffy. Once they're ready, bake for 30 minutes.
Allow to rest for 5 minutes before icing.