Warm the milk and butter in the microwave until the milk is warm and the butter is mostly melted. Transfer to a large mixing bowl, whisk in the brown sugar and yeast, and let sit until slightly foamy, about 5 minutes.

Add the flour, salt, and egg and mix with a dough hook until combined, then mix in the shredded carrots until evenly incorporated. Cover the bowl and let the dough rise until doubled in size, about 1 hour.

Turn the dough out onto a floured surface and roll into a 16×14-inch rectangle. Spread the softened butter evenly over the dough, then sprinkle with the combined brown sugar, cinnamon, nutmeg, and ginger. Roll tightly from the long side into a log and cut into 12 equal pieces.

Place the rolls in a greased 9×13-inch baking dish, cover, and let rise in a warm spot for at least 2 hours. Preheat the oven to 350°F, then bake uncovered for 30–35 minutes, or until golden brown.

While the rolls cool slightly, beat the cream cheese, butter, and powdered sugar until smooth. Spread the frosting over the warm rolls and top with walnuts and candy eggs if desired. Store leftovers covered in the refrigerator.