Place 1/2 cup warm milk in a large mixing bowl. Add 1/2 teaspoon of the sugar to the milk and stir. Sprinkle the yeast over the mixture and stir until dissolved. Allow the mixture to sit for around 15 minutes to allow the yeast to bloom. 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 the dough for about 5 minutes at LOW speed. Cut butter into four pieces and add to the dough, one at a time, mixing with the dough hook between each addition. Turn the speed to medium and knead for about 5 more minutes. The dough should come together and form a ball.
Place the dough in a greased/oiled bowl, turning once oil the top. Cover with plastic wrap and allow it to rise in a warm place until doubled, approximately 1 to 1.5 hours.
Divide dough into two equal parts and roll each half to an 8 x 20 rectangle (approximately).
Stir softened butter and maple extract until combined. Spread over the surface of the rectangles.
Combine cinnamon and sugar and sprinkle half over each buttered rectangle.
Starting at the long end, roll each rectangle into a log, gently pulling backward during the process to help the roll stay tight. Pinch the seams together to seal. Cut into 2-inch slices and place in greased baking pans. Cover the pans and let rise until doubled in a warm place, about an hour. Preheat oven to 350 degrees Fahrenheit.Brush the tops of the risen rolls with the reserved 1/4 cup of milk. Bake on the center rack of the oven for 30-35 minutes or until golden brown. Remove from the oven and allow to cool in the pan.
Spoon or pour the icing over the tops of the rolls while still slightly warm.