Combine the flour, yeast, sugar, and salt in a large bowl. Add the warm milk, melted butter, and egg, then mix until a soft dough forms.
Knead the dough by hand or with a mixer until it becomes smooth and elastic. If it feels sticky, add a little more flour, one spoonful at a time.
Place the dough in a lightly greased bowl, cover it, and let it rise until doubled in size. This usually takes about 1 hour, depending on the warmth of your kitchen.
Roll the dough into a rectangle. Brush with melted butter, then sprinkle evenly with the cinnamon and brown sugar mixture.
Roll the dough up tightly from the long side. Slice it into even pieces and arrange the rolls in a greased skillet, leaving a little space between them.
Cover the skillet and let the rolls puff slightly. Bake until golden and fragrant, with the centers baked through and soft.
Whisk the powdered sugar, milk or cream, and alcohol-free vanilla until smooth. Drizzle the glaze over the warm rolls and serve right away.