Heat the milk in a small saucepan until it bubbles and then remove from heat. Mix in the butter until it has melted and let it cool to lukewarm.
Next, in the bowl of a mixer, mix 2 1/4 cups flour, yeast, salt, and sugar. Add water, egg, and the milk and beat well. Next, add the remaining flour a little bit at a time (stirring well after each addition). Once the dough has pulled away from the sides of the bowl, knead it on a lightly floured surface about five minutes.
Next, cover the dough with a damp cloth for ten minutes.
Mix brown sugar, butter, and cinnamon together for the filing.
Roll out dough into a rectangle (about 1/4 inch thick). Spread the filling mixture over it. Roll the dough and pinch the seams to seal it. Cut into 12 equal sized rolls and place in a circular pan (with space in between them so about 6 per pan) and place a damp cloth over them until they double in size. I generally preheat the oven to 200 degrees, put the cinnamon rolls in, and then turn off the oven. There the rolls have a warm place to rise, and with the instant yeast, it takes about 30 minutes
Preheat the oven to 375 degrees and bake until golden brown (about 12-15 minutes).
While the rolls are cooking, beat the butter for the frosting 'til creamy. Then add the maple syrup, powdered sugar, vanilla, and salt and beat until well mixed. Then, add the milk, a little at a time, until you have the desired consistency.
When the rolls are done, spread the icing over and allow them enough time to cool so that you don't burn your mouth, but do eat them warm because it's worth the sticky fingers.