In a large bowl, whisk together 2 cups flour, 2 tablespoons sugar, the yeast, and salt. Pour in warm milk, melted butter, and egg. Stir until a shaggy dough forms, then knead on a floured surface about 5 minutes until smooth and elastic.

Roll the dough into a 12x8-inch rectangle. Spread the 3 tablespoons softened butter over it, leaving a 1/2-inch border along one long edge. Sprinkle brown sugar and cinnamon evenly, then press gently.

Roll the dough up tightly from the long edge, pinch the seam, and cut into 6 equal rolls. Butter an 8-inch pan, arrange rolls cut side up, cover loosely, and let rise until puffy and nearly doubled, 45 to 60 minutes.

Preheat oven to 350°F. Bake rolls 22 to 25 minutes until golden and centers reach 190°F. Cool in the pan 5 minutes.

Whisk 1/2 cup powdered sugar, 1 tablespoon milk, and vanilla. Add more milk as needed for a thick, pourable glaze. Drizzle over warm rolls and serve.