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.