For the sweet rolls:

• Remove the peel from 1 lemon using a vegetable peeler (try not to get the bitter white pith under the peel.)  Place the lemon peel and milk in a small saucepan.  Heat the milk until it's a little steamy, but do not boil!  Set aside to steep.
• 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 sugar, ⅓ cup melted butter, salt and the egg.  Strain the milk into the bowl removing and discarding the lemon peel.  Blend the mixture by hand using a spoon or spatula until combined.  Add two cups of the flour and turn the mixer on low.  Blend until all the flour is incorporated.  Add the yeast/water mixture and continue mixing until incorporated.  Add 1 more cup of flour and continue to mix until blended.  Finally, add the remaining ½ cup of flour to make a soft dough that's just a little sticky.  Turn the mixer on medium-low and knead for 5 minutes or until you have a smooth dough.  Add a tablespoon of flour if needed to help clean the sides of the bowl.  Don't add too much flour or the rolls will be dry.
• Place the dough in a clean, well 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 ½ hours.
• While the dough is rising, prepare the filling.  Combine ⅓ cup of softened butter, ⅓ cup of granulated sugar and the juice and zest from 1 lemon.  Mix until well blended.  Set aside.
• Grease or butter two 8 or 9-inch round cake pans or one 13x9-inch baking pan.  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 15 by 10-inch rectangle.  Spread the filling all over the dough leaving a ½-inch border on the 15-inch side.  Roll the dough up beginning on the 15-inch side with filling.  Roll the dough somewhat tight, tucking and rolling as you go.  Lightly dampen the ½-inch edge with water using your finger.  Pinch the seam to seal. Cut the roll in half, then 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 to 60 minutes or until doubled in size and puffy.  Preheat oven to 350℉.
• Bake the rolls for 25 to 30 minutes or until nicely browned.
• Mix the icing ingredients together and spread on warm rolls.  Serve immediately.  Refrigerate leftover rolls.