USING ACTIVE DRY YEAST: Heat 1/4 cup water to 105°F-115°F in microwave at 15-second intervals. Use a kitchen themometer to ensure the correct temperature.Stir and dissolve active yeast in the water. Stir in 1 tsp granulated sugar. Allow yeast to activate; about 5 minutes. Yeast should foam/froth and double in size, if not see notes below.
USING INSTANT YEAST: Skip above step and continue below.
Heat whole milk in microwave for 30-45 seconds. Add butter, ¼ cup granulated sugar, and ½ tsp salt. Stir to dissolve. Cool until warm to touch.
In a stand mixer fitted with a dough hook (or by hand) pour in milk mixture.
Turn mixer on medium-low. Add egg and yeast (actively proofed or instant) and mix until combined.
Add ½ cup flour at a time to bowl. Slowly add flour until dough forms and starts to pull away from the sides of the bowl. If the dough is incredibly sticky, add a little more flour. Dough should only lightly stick to fingers. NOTE: Active dry yeast usually uses closer to 4 cups flour due to extra proofing liquid while instant yeast will use closer to 3.5 cups flour.Turn off mixer and lift dough hook. Use a closed fist to knead the dough a few times. Lightly grease a large bowl with oil. Place dough in bowl, move it around then flip it over to ensure it's lightly covered in oil. Cover tightly with plastic wrap or a kitchen towel.Place in a warm, draft-free location with a house temperature of 80°F-90°F. See notes for easy solutions to a cold house.Allow dough to double in size; about 1 hour.
Mix together brown sugar, cinnamon, and flour; set aside.
Lightly dust work surface with flour. Take dough out of bowl and knead a few times into a ball. Place in center of work surface.
Using a rolling pin or your hands, roll/pat the dough into a 14X10-inch rectangle. It's okay to roll the dough past this size. Use hands to reshape by patting the outer dough inward into a 14X10-inch rectangle.
Smear soft butter over dough. Evenly spread then pat cinnamon sugar filling into butter.
Roll the dough up tightly without pulling it. Start at one long end and roll the dough to the other side. Pinch together at the seam.
Use a serrated knife to cut dough roll into 12 equal rolls. Make first cut in the center. Then cut those to halves in half again. Then, cut those halves into 3rds.
Place individual rolls in a buttered 9X13-inch casserole dish. Cover with plastic wrap and place in a warm location until dough is double in size; about 1-1.5 hours.
To make the maple frosting, cube cold butter, and cream cheese. Beat on high with mixer until combined. Beat in maple flavoring, salt, milk, and powdered sugar and mix until fully combined and smooth.
Preheat oven to 350°F. Bake rolls for 18-28 minutes. At the 18-minute mark, check rolls. Watch carefully and bake until only a few very very light golden spots appear on crust super. Do NOT bake until fully golden.
Allow rolls to cool; 15 minutes. Spread maple icing evenly over rolls.