Mix 2 cups flour, the sugar, yeast, and salt. Stir in the milk, melted butter, and egg until a shaggy dough forms. Knead 8-10 minutes, adding remaining flour gradually, until the dough is smooth and tacky.

Roll the dough into a 10x16-inch rectangle. Spread the softened butter over the surface, then sprinkle with the brown sugar and cinnamon mixture.

Cut the rectangle lengthwise into 6 strips. Stack them in two piles of 3, then cut crosswise into 6 equal stacks. Place the stacks cut sides up in a greased 9x5-inch loaf pan.

Cover and let rise in a warm place until puffy and nearly doubled, about 45-60 minutes.

Preheat oven to 350°F (175°C). Bake the loaf for 30-35 minutes, until deep golden on top and 190°F (88°C) internal.

Beat cream cheese and butter until smooth, then mix in powdered sugar and vanilla. Add milk slowly until the glaze is thick but pourable. Pour over the warm loaf and serve immediately.