Warm milk and butter in a cup in the microwave for 45 seconds. Mix in yeast and 1/4 cup of sugar and let sit for 5-10 minutes until it looks foamy. When yeast is ready, pour into stand mixer and add in the rest of the sugar, and vanilla. Mix together.
Using the dough hook, add in flour on slow speed. Mix with dough hook until the dough has formed. Add in a couple more tablespoons of flour if the dough looks really sticky and isn't forming into a ball.
On a lightly floured surface, knead the dough for about 5 minutes until it feels smooth. Place in a lightly greased bowl and cover with a damp towel. Let rise in a warm dry place for one hour.
While dough is rising, combine the brown sugar and cinnamon together in a small bowl. Then add softened butter and smush together using a spoon until it's smooth and incorporated.
When the dough is ready, punch down and put back onto your floured surface. Roll out into a 12" x 24" rectangle. Spread cinnamon filling onto the dough, leaving a small untouched border around the edges. Roll on the long side tightly. Cut into 12 equal pieces. They should be about 2" wide.
Place 6 buns each evenly spaced out into two 9" x 13" tins/pans. Set oven to preheat at 350°F. While the oven is preheating, let the buns rise one final time for a half hour. Bake about 15-20 minutes or until they are golden brown.
While the buns are baking, make the icing. Beat the butter until fluffy and beat in vanilla. Then beat in powdered sugar a half cup at a time. Place onto buns once they come out of the oven. Enjoy them hot!