Place ½ cup warm milk in a large mixing bowl. Add ½ tsp. of the sugar to the milk and stir. Sprinkle the yeast over the mixture and stir until dissolved. Allow mixture to sit for approx. 15 minutes to allow the yeast to activate. The mixture will be bubbly when it is ready.
Once the yeast mixture is ready, add the starter to it along with the remainder of the sugar, egg, flour and salt. Mix by hand until combined.
Using a dough hook, knead dough for about 5-minutes on low speed. Cut butter into four pieces and add to dough, one at a time, mixing with the dough hook between each addition. Turn the speed to medium and knead for about five more minutes. Dough should come together and begin to form a ball.
Place dough ball in a greased bowl, turning once to grease top. Cover with plastic wrap and allow to rise in a warm place until doubled (about 1-1½ hours,
Divide dough in two equal parts and roll each half to an approx. 8x20 inch rectangle.
Stir softened butter and maple extract until combined. Spread over surface of rectangle,
Combine cinnamon and sugar and sprinkle half over each buttered rectangle. Using your rolling pin, lightly roll across surface to press mixture into dough.
Starting at one long end, roll up into a log, gently pulling backwards during the process to help the roll to stay tight. Pinch seam together to seal.
Cut into 2-inch slices and place in a greased baking pan.
Repeat process with second half of dough.
Cover pans and let rise until doubled in a warm place.
Preheat oven to 350°.
Brush tops of risen rolls with the reserved ¼ cup of milk. Bake on center rack of oven for about 35 minutes, until golden brown. Cool in pan. until warm.
Ladle or pour maple icing over tops of rolls while still warm.