In a small sauce pan over medium low heat, melt the butter until it just starts to bubble. Remove from heat and stir in milk and sugar.
Pour the butter, milk and sugar mix into the bowl of stand mixer or a large mixing bowl. Sprinkle the yeast over the top and let sit for 5 minutes.
With the dough hook attachment or a wooden spoon/rubber spatula, stir in the salt and 1 cup of flour at a time on low speed. To test if you have enough flour pinch the dough with your fingers, the dough should be smooth and a tad tacky, but it should not stick to your fingers when pinched. If you need more flour add it 2 tablespoons at a time.
Bump up the speed to medium and knead the dough for 1 minute, or knead by hand until smooth.
Cover bowl with a clean towel or plastic wrap and place in a warm, draft-free spot for 30 minutes or until doubled in size.
Once its doubled, dump it onto a lightly flour surface and roll it out into a 15x10 inch rectangle that is about ½ inch thick.
Brush the dough with the melted butter leaving a 1 inch space on one long edge for sealing.
Mix together the sugar, brown sugar and cinnamon. Sprinkle it evenly over the melted butter.
Roll the dough tightly starting from the long edge. Pull the end up and lightly pinch to seal the dough.
Cut the dough into 12 rolls using a dough cutter or serrated knife. (Cut in half, then cut the halves in half to create 4 pieces. Cut each quarter into thirds.)
Place rolls in a greased 9x13 baking dish, cover and place in warm area again for 30 minutes.
Preheat oven to 375°F. Bake rolls for 15-20 minutes until golden brown and center is no longer doughy.
While the rolls bake, make the frosting. Beat together the butter and 1 cup of powdered sugar until smooth.
Add the vanilla extract, remaining powdered sugar and milk. Beat on low until fully combined. Then bump speed to medium and beat until light and fluffy.
Dollop each roll with frosting then spread it out with an off set spatula.