In a small bowl, combine the milk, yeast, and 1 tbsp. of the sugar. Let stand for 5 minutes until the yeast foams. Whisk in the salt and egg. Set aside.
Measure the flour into a large mixing bowl or into the work bowl of a food processor. Cut the butter into the flour until the mixture resembles coarse crumbs.
Pour the liquid ingredients over the flour mixture. Mix lightly with a fork or use short pulses if using the food processor, until the flour is just moistened. Cover with plastic wrap and refrigerate at least 4 hours or up to 2 days.
When you’re ready to shape the cake (you can shape the coffee cake, cover, refrigerate overnight, and bake it in the morning; or you can shape it, let it rise, and then bake it), remove the dough from the refrigerator.
Lightly grease a baking sheet or cover with parchment paper.
Turn the dough out onto a lightly floured surface and roll it out to make a 15 in. circle about 1/3 in. thick. To fill the cake, brush with the melted butter. Mix the sugar and cinnamon in a small bowl and sprinkle over the dough. Sprinkle with the nuts.
Roll up jellyroll fashion. Place on the prepared baking sheet. Fold the roll in half so that one half is on top of the other. Seal the open ends together. With a sharp knife, make a cut from the folded end lengthwise down the center of the roll, cutting from the top roll through to the bottom to within 2 in. of the sealed end. In so doing the folded end of the roll will separate out to form the two lobes of a heart shape. Spread out the two sides to expose the filling.
Let rise, lightly covered, until puffy, about 45 minutes, or cover and refrigerate over night. If refrigerated, remove from the refrigerator 1 hour before baking so that the coffee cake can rise and come to room temperature.
Preheat the oven to 375 degrees. Bake for 20 to 25 minutes until light brown. Remove and cool on a wire rack.
In a small bowl, combine the icing ingredients. Drizzle over the warm coffee cake.