Make the dough: Whisk the flour, sugar, salt, and yeast together. Set aside. Heat the milk and butter together in the microwave until the butter is melted and the mixture is warm to the touch, between 100-110°F, 38-43°C. Pour the warm milk mixture into the dry ingredients. Add the egg. Using a handheld or stand mixer fitted with a dough hook or paddle attachment, or using a rubber spatula and mixing by hand, beat/mix the mixture until a soft dough forms. The dough is ready when it gently pulls away from the sides of the bowl. If the dough is pretty sticky, beat/mix in 2-4 extra tablespoons of flour.
Turn the dough out onto a lightly floured surface. With lightly floured hands, knead the dough for 3 minutes. If you'd rather use the mixer to knead it, beat the dough on low speed for 3 minutes instead. After kneading, poke the dough with your finger – if it slowly bounces back, the dough is ready for the next step. If not, keep kneading.
Form the dough into a ball on your work surface. Cover it with a clean towel, aluminum foil, or plastic wrap, then let it rest for 30 minutes at room temperature.
Make the filling: After 30 minutes, use a rolling pin to roll the dough out into a 15x12-inch rectangle. Evenly spread the softened butter on top. Mix the sugar and cinnamon together and sprinkle it all over the buttered dough.
Grease a 9-inch round pie dish or cake pan with butter or nonstick spray.
Shape the cinnamon roll cake: Using a very sharp knife or a pizza cutter, cut the dough into six 2-inch wide strips. Roll up one strip and place it in the center of the pan. Coil the remaining 5 strips around the center roll, starting each strip at the end of the previous one to make one large cinnamon roll. Some of the filling may spill out as you pick up the strips, so just sprinkle it on top of the cake when you're done shaping it.
Loosely cover the cake with aluminum foil or plastic wrap and allow it to rise in a relatively warm, draft-free environment for 1-2 hours or until double in size.
Bake: Preheat the oven to 350°F (177°C). Bake for 30-35 minutes or until golden brown. If you notice the top or edges browning too quickly, tent the cake with aluminum foil. While the giant cinnamon roll cake is baking, some spots may rise up more than other spots. That's completely normal and expected from something this large. Simply remove the pan from the oven and, with the back of a spatula, press down the spots that are rising taller than others to help keep them even. Return to baking.
Remove the finished cinnamon roll cake from the oven and cool in the pan on a wire rack for 10 minutes.
Make the icing: Whisk the icing ingredients together. If you want a thicker icing, whisk in more confectioners' sugar. If you want a thinner icing, whisk in more milk or cream. Drizzle all over warm cake and serve.
The cinnamon roll cake is best enjoyed the same day it's made, but will remain fresh covered tightly for up to 4 days in the refrigerator.