Warm the milk and stir in 1 tsp sugar and the yeast; let sit until foamy, then mix in butter, egg, and remaining sugar. Add 3 cups flour and salt, knead until smooth, adding remaining flour as needed.

Let dough rise until doubled, then punch down. Roll into a 12x16-inch rectangle, spread with soft butter, sprinkle with brown sugar and cinnamon, and roll up tightly from the long side.

Cut into 12 rolls, place in a greased 9x13-inch pan, cover, and let rise until puffy, 30–45 minutes.

Bake at 350°F (180°C) for 20–25 minutes, until golden and set.

Beat cream cheese, butter, powdered sugar, and vanilla until smooth, thinning with milk if needed. Spread over warm rolls and serve.