Preheat the oven to 375°F (190°C). Line a large baking sheet (about 13x18 inches) with parchment paper.
Activate the yeast: in a large bowl, whisk together the warm milk, granulated sugar, and yeast. Let stand for about 5 minutes, until foamy.
Add the melted butter, egg, and salt. Mix well, then gradually stir in the flour until a soft dough forms.
Knead the dough on a lightly floured surface (or in a stand mixer with a dough hook) until smooth and elastic, about 5–8 minutes.
Place the dough in a greased bowl, cover, and let rise in a warm spot until doubled in size, about 60 minutes.
Turn the dough out onto a floured surface and roll into a large rectangle about 1/4-inch thick.
Spread the softened butter over the dough. In a small bowl, mix brown sugar, cinnamon, and nutmeg (if using), then sprinkle evenly over the butter.
Cut the dough into strips that are wider at the bottom and narrower at the top (so the rolls create a tree shape). Roll each strip up tightly to form individual cinnamon rolls.
Arrange the rolls on the prepared baking sheet in a Christmas tree shape (for example: 4 rolls on the bottom row, then 3, then 2, then 1 at the top). Add one roll at the bottom as a trunk if desired. Leave a little space for expansion.
Cover and let the shaped rolls rise for about 20 minutes, until slightly puffy.
Bake for 18–22 minutes, until golden brown and cooked through.
Meanwhile, make the icing: beat cream cheese and softened butter until smooth. Mix in powdered sugar and vanilla, then add milk 1 tablespoon at a time until you reach a drizzleable consistency.
Drizzle or spread the icing over the warm rolls. Serve immediately while soft and gooey.