Make the dough: In a large bowl, combine red velvet cake mix, 1 cup flour, and yeast. Add warm water and beat on medium speed for 2 minutes. Stir in enough remaining flour to form a soft, sticky dough.
Knead and rise: Turn dough onto a lightly floured surface and knead gently 6–8 times. Place in a greased bowl, turning once to coat. Cover and let rise in a warm spot until doubled, about 2 hours.
Prepare filling: In a small bowl, mix brown sugar and cinnamon.
Shape rolls: Punch down dough and roll into an 18x10-inch rectangle. Brush with melted butter, leaving a 1/4-inch border. Sprinkle evenly with cinnamon-sugar mixture. Roll up tightly, jelly-roll style, starting from a long side. Pinch seam to seal.
Cut and rise again: Cut into 12 slices and place cut-side up in a greased 13x9-inch baking pan. Cover with a kitchen towel and let rise until nearly doubled, about 1 hour.
Bake: Preheat oven to 350°F (175°C). Bake for 15–20 minutes, until puffed and lightly browned. Cool slightly.
Make icing: Beat confectioners’ sugar, butter, vanilla, and enough milk to reach a drizzling consistency.
Finish: Drizzle icing generously over warm rolls and serve.