Warm the milk slightly, just until it's slightly above room temperature. Add it to a large liquid measuring cup.
Add the sugar and the yeast and whisk with a fork to combine. Set aside for a few minutes until the yeast is bubbly.
Into the bowl of your stand mixer, measure out the 3 1/2 cups all purpose flour and the salt. Mix these two together for a few seconds, just until they're combined.
Add the egg and the melted butter to the flour and then add the milk mixture with the sugar and yeast.
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.
Cover the bowl with some plastic wrap and move it to a warm place in your kitchen to rise for about 1.5 to 2 hours. It should double in size.
Once you notice your dough has risen fully and before you do anything with it, grease a large rectangular baking pan or ceramic baking dish (approximately 9x13 inches in size) with some butter. Pour the maple syrup into the bottom of 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 oil. Using a rolling pin, roll out the dough evenly until you have a rectangle that's approximately 18 inches wide and 8 to 10 inches deep. It's important that the dough is wider than it is deep so the rolls have enough height when they're baked.
Brush the dough rectangle with the melted butter.
In a separate bowl, combine the brown sugar and the cinnamon. Spread the brown sugar and cinnamon mixture evenly over the dough.
Roll up the dough from long side to long side until you've got one long roll. Cut the long roll in half, then each half 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 because they're different sizes.
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.
Preheat your oven to 350 degrees Fahrenheit (175 degrees Celsius)
Once the rolls have risen for 20 minutes (don't let them rise longer than this), 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 25-30 minutes or until they're a nice even golden brown. The egg wash is an optional step - it helps the tops of the rolls brown, but it's not a necessary step.
Remove the pan from the oven and let the buns cool in the pan for about 20-30 minutes before cutting into them.