In a small saucepan, whisk together the flour and milk until smooth. Cook over medium heat, whisking constantly, until thickened into a paste that leaves visible lines when stirred, about 2–4 minutes. Transfer to a bowl and let cool to room temperature.

In a small bowl, whisk the warm milk and sugar until dissolved. Sprinkle the yeast over the top and let sit for 5 minutes until foamy.

In the bowl of a stand mixer fitted with the dough hook, combine the cooled tangzhong, yeast mixture, melted butter, vanilla, eggs, salt, cinnamon, nutmeg, shredded carrots, and both flours. Mix on low speed until a soft dough forms.

Knead on medium speed for 5–8 minutes until smooth and elastic. Add the softened butter one tablespoon at a time, mixing until fully incorporated. The dough will be soft and slightly sticky.

Transfer the dough to a greased bowl, cover, and let rise in a warm place for 1–2 hours, or until doubled in size.

For easier shaping, refrigerate the dough for 2–8 hours or overnight after the first rise.

In a medium bowl, mix together the softened butter, brown sugar, cinnamon, ginger, nutmeg, and salt until smooth.

Fold in the grated carrots and chopped walnuts until evenly combined. Set aside.

Grease a 9×13-inch baking pan.

Roll the chilled dough into a 16×12-inch rectangle with the long side facing you. Spread the carrot cake filling evenly over the dough, leaving a ½-inch border along the far long edge.

Starting from the long edge closest to you, roll the dough tightly into a log. Slice into 12 even rolls using a sharp knife or dental floss.

Arrange the rolls in the prepared pan. Cover loosely and let rise for 45–60 minutes, until puffy and nearly doubled.

Preheat the oven to 325°F (165°C). Bake the rolls for 28–32 minutes, until set and lightly golden on top. Tent loosely with foil if the tops brown too quickly.

Let the rolls cool for about 10 minutes before frosting.

In a small saucepan, melt the butter over medium heat. Continue cooking until the butter foams and turns golden brown with a nutty aroma. Remove from heat and let cool slightly.

In a bowl, beat the softened cream cheese until smooth. Add the brown butter, powdered sugar, vanilla, and salt, mixing until creamy and spreadable. Depending on the cream cheese brand you use you might need to add a splash of cream or milk to get the right texture frosting.

Spread the frosting over the warm (not hot) rolls.