Brown the butter: Brown 85g butter for the dough, 113g for the filling, and 50g for the frosting. Cook until golden and nutty-smelling, then transfer to separate bowls to cool.
Make the dough: Mix flour, sugar, yeast, and salt. Add warm milk, egg, and cooled brown butter. Knead for 8–10 minutes until smooth and elastic. Add a little extra milk only if the dough feels too dry.
First rise: Place dough in a greased bowl, cover, and let rise for 1–1½ hours, until doubled.
Make the filling: Mix cooled brown butter with brown sugar, granulated sugar, cinnamon, vanilla, and salt until smooth and spreadable.
Roll: Roll the dough into a large rectangle and spread the cinnamon filling evenly over the surface.
Shape: Roll tightly from the long side into a log. Cut into 10–12 equal rolls.
Second rise: Place rolls in a greased or parchment-lined baking dish. Cover and let rise for 30–45 minutes, until puffy.
Bake: Preheat oven to 350°F (175°C). Bake for 20–30 minutes, until golden brown and cooked through.
Make frosting: Beat cream cheese until smooth. Add cooled brown butter and mix. Gradually add powdered sugar, vanilla, and salt. Add heavy cream until the frosting reaches your preferred consistency.
Frost: Cool the rolls for 5–10 minutes, then generously spread with brown butter cream cheese frosting.
Serve: Serve warm and enjoy.