In a mixing bowl, combine the flour, sugar, yeast, and salt. Mix well.
In a separate bowl, whisk together the warmed milk, melted butter, and egg.
Pour the wet ingredients into the dry ingredients and mix until a dough forms.
Knead the dough on a floured surface for about 5-7 minutes until smooth and elastic.
Place the dough in a greased bowl, cover with a towel, and let it rise in a warm place for about 1 hour or until doubled in size.
Preheat the oven to 350°F (175°C).
Roll out the dough on a floured surface into a large rectangle.
Spread the softened butter over the dough, then sprinkle with brown sugar and cinnamon.
Roll the dough tightly from one long side to the other, forming a log.
Cut the log into 12 equal pieces.
Arrange the pieces in a circular pattern on a greased baking sheet to form a snowflake shape.
Cover and let rise for another 30 minutes.
Bake in the preheated oven for 20-25 minutes, or until golden brown.
While the rolls are baking, prepare the icing by mixing the powdered sugar, milk, and vanilla extract until smooth.
Drizzle the icing over the warm cinnamon rolls before serving.