Cube 1-½ cups unsalted butter and place in a light-colored skillet. Melt the butter over medium heat, stirring constantly. After 5-8 minutes, the butter will begin browning; remove from heat and freeze until solidified, about 20 minutes.
In a mixing bowl, combine 4 cups of flour, yeast, sugar, and salt. Add the milk, brown butter, eggs, and vanilla bean paste and mix until a shaggy dough forms. Add the remaining ¼ cup of flour and beat until a soft dough forms. Knead the dough for 1 minute and transfer to a greased bowl. Cover and let rest for 15 minutes.
For the filling, combine brown butter, dark brown sugar, and cinnamon in a bowl until a thick paste forms. Grease a baking dish and roll out the dough into a rectangle. Spread the filling over the dough, roll it up, and cut into 12 rolls. Arrange in the baking pan and let rise for 50-60 minutes.
Preheat the oven to 350°F. Bake the rolls for 20-25 minutes until golden brown.
For the icing, whisk together cream cheese and brown butter, then add powdered sugar, vanilla bean paste, and milk until smooth. Drizzle over warm rolls.