In the bowl of a stand mixer, combine the milk, maple sugar, and yeast. Let sit until foamy, about 5-10 minutes. Whisk in the butter and beaten egg.
With the mixer running on low, add 3 cups of flour. Continue adding flour 1 tablespoon at a time until a soft, tacky, but cohesive dough forms and pulls away from the sides of the bowl. Continue kneading for 10-15 minutes.
Grease a clean bowl with oil. Flip the dough into the bowl so both sides are coated. Cover in plastic wrap and let rise in a warm, draft-free spot until doubled, about 60-90 minutes.
Lightly grease a 9 x 13 inch casserole dish. Flour a clean, flat surface. Punch down the dough and dump onto the floured area. Roll into a roughly 12 x 18 inch rectangle. Spread with butter, then sprinkle with the lemon sugar. Roll from the long side into a log. Use a sharp knife to slice 12 rolls out of the dough. Arrange into the prepared baking dish. Cover and let rest 20-30 minutes longer until puffy. Or, transfer to the fridge and let rise overnight.
Preheat oven to 375 degrees. Uncover the rolls and bake for 18-20 minutes until golden on top. If you refrigerated the rolls, you may need 22-25 minutes. Let cool 10-15 minutes.
In a large bowl, beat the butter and cream cheese on medium speed until combined, then whip in the lemon juice and vanilla. Add powdered sugar 1/2 cup at a time, mixing between each addition until incorporated. Crank up the speed to high and whip until very fluffy, about 2-3 minutes. Spread generously onto the warm rolls. ENJOY!