Into the bowl of your stand mixer, measure out the 3 1/2 cups all purpose flour and the yeast. Mix these two together for a few seconds, just until they're combined.
Next, melt 1/3 cup butter in a saucepan on the stove over medium heat. If you're thinking to cut a corner here and melt your butter in the microwave instead, don't do it! If the liquid is too hot, it will kill the yeast. Once it's melted and starts to bubble, remove it from the heat and stir in the 1/3 cup sugar and the salt until you can see and feel the grains beginning to dissolve in the hot butter. Add the milk slowly, stirring constantly. You're looking for the milk and butter mixture to be about 105 degrees Fahrenheit, but if it feels only slightly warm to the touch (slightly above room temperature) you're good to go.
When your wet ingredients have reached the ideal temperature, add in an egg and break it up with a fork into the mixture.
Pour the wet ingredients into the dry ingredients in the bowl of a stand mixer. Make sure your stand mixer is fitted with the dough hook attachment.
Turn your mixer on low and watch the ingredients combine. After about a minute you should see a sticky dough starting to come together. If you need to turn the mixer off and scrape down the sides and bottom of the bowl, now's your chance.
Turn up the mixer speed slightly and allow the mixture to knead the dough for about 3 minutes. The dough should be tacky to the touch, but not so sticky that it's very messy. It should pull away from the sides of the bowl easily and the bowl should appear fairly clean. If the dough appears too sticky, continue kneading and add flour, one tablespoon at a time, until the dough reaches that tacky texture.
If you don't have a stand mixer, you can knead the dough by hand on a lightly floured work surface for 5 minutes after the wet and dry ingredients are combined. If you're at all serious about cooking and baking, a stand mixer is a great investment. I have had my KitchenAid stand mixer
for 10 years and it has paid for itself a few times over I'm sure!
Cover the bowl with some plastic wrap and move it to a warm place in your kitchen to rise for about 2 hours. It should have doubled in size after about 2 hours.
Once you notice your dough has risen fully, and before you do anything with it, grease a large rectangular pan (at least 9x13 in size) with some butter. Pour the maple syrup into the pan, tilting the pan if necessary to make sure the syrup coats the bottom completely.
Turn the dough out onto a work surface that's been very lightly greased with vegetable oil. Using a rolling pin, roll out the dough evenly until you have a rectangle that's approximately 18 inches wide and 12 inches deep. Brush the dough with the melted butter.
In a separate bowl, combine the brown sugar and the cinnamon. I use 1 tbsp of cinnamon, but feel free to use a little less if you'd prefer a more subtle taste. Spread the brown sugar and cinnamon mixture evenly over the dough.
Roll up the dough from the long side until you've got one long roll. Cut the long roll in half, then in half again, and then each of those sections into 3 pieces until you have 12 cinnamon rolls. Try to cut the rolls to an even thickness as this promotes even baking. You don't want some rolls to be overdone and others to be raw in the middle.
Arrange the 12 pieces in the pan you've greased and nestle them into the maple syrup in the bottom. An arrangement of 3 rows of 4 works well in a rectangular pan.
Cover the pan with a clean kitchen towel and set them aside to rise. It's a good idea to preheat your oven at this time, too. Set temperature to 350 degrees Fahrenheit.
Once the rolls have risen for 20 minutes (no longer), very lightly brush the tops with some egg wash (beat one egg with 1 tablespoon of water) and place them in the oven to bake for about 40 minutes, or until they're a nice even golden brown.
Remove the pan from the oven and let the buns cool in the pan for about 15 minutes before cutting into them.