Mix lukewarm milk, yeast, sugar, egg, yolk, and cooled melted butter. Add 3 1/4 cups flour and salt, stirring until a shaggy dough forms.
Knead 8–10 minutes by hand or 6–8 minutes with a dough hook until smooth and slightly tacky. Add reserved flour sparingly, only if the dough remains unmanageably sticky.
Place in a lightly greased bowl, cover, and rise 60–90 minutes in a warm draft-free place until roughly doubled. Timing depends on room temperature.
Roll into a 12 x 16-inch rectangle on a lightly floured surface. Spread filling butter evenly and scatter mixed brown sugar and cinnamon over it.
Roll from a long edge into a log and cut into twelve equal pieces with unflavored dental floss or a sharp knife. Arrange in the greased baking dish.
Cover and rise 30–45 minutes until puffy and touching. Heat the oven to 350°F (175°C) near the end of this second rise.
Bake 24–30 minutes until golden and the center rolls are cooked through, around 190°F (88°C) internally. Cool 15 minutes before icing.
Whisk powdered sugar, milk, and vanilla into a thick drizzle. Spread over warm, not piping-hot rolls, allowing it to settle before serving.
