In a small bowl dissolve the yeast in the warm water. Set aside for 15 minutes.
In the bowl of a stand mixer fitted with a dough hook, combine the milk, sugar, melted butter, salt and egg. Blend the mixture by hand using a spatula, until combined. Add two cups of flour and turn the mixer on low. Blend until all the flour is incorporated. Add the yeast/water mixture and continue mixing on medium-low until incorporated. Add 1 more cup of flour and mix until blended.
Finally add the remaining 1/2 cup of flour to make a soft dough that is just a little sticky. Turn the mixer on medium-low and knead for 5 minutes or until you have a smooth dough. Add one tablespoon of flour if needed to help clean the sides of the bowl toward the end of mixing. Don’t add too much flour as the rolls will be dry.
Place the dough in a clean, wet oiled bowl and turn to coat the dough. Cover the bowl with plastic wrap and move to a warm location free from drafts. (I use my microwave as a proofing oven with the task light on for warmth.) Allow the dough to rise until doubled in size, about 1 to 1.5 hours.
Grease two 8 or 9-inch round baking pans, or a 13x9-inch pan. Sprinkle the bottom of the pan with 2 teaspoons coarse sugar. Set aside.
When the dough has doubled in size, gently punch it down and turn out onto a clean, lightly floured work surface. Roll the dough into a 15x10-inch rectangle. Spread the jam all over the dough leaving a 1-inch border on the 15-inch side farthest away from you. Sprinkle the chopped strawberries evenly over the strawberry jam.
Roll the dough up beginning on the 15-inch side with filling (closest to yo- rolling towards the side with no jam or berries along the edge. Roll the dough somewhat tight, tucking and pulling slightly to create a cylinder. Lightly dampen the 1-inch border with water using your finger. Pull the dough up and pinch to seal the seam.
Cut the cylinder in half, then cut each half in half again. Cut each one-forth section into 3 equal size rolls. Place the rolls cut side up in the prepared pan(s).
Move the rolls to a warm location free from drafts and allow them to rise for 45-60 minutes or until doubled in size and puffy.
30 minutes before baking, preheat oven to 350F.
Bake the rolls for 20 to 25 minutes or until nicely browned.
Mix the icing ingredients together and drizzle on the warm rolls. Serve immediately.