In a small bowl, dissolve yeast in warm water and set aside until frothy.
In a large bowl, combine sugar, melted butter, milk, and salt. Stir in yeast mixture and beaten eggs.
Add grated carrots and gradually stir in flour to form a soft dough.
Knead the dough for about 8 minutes on a floured surface, until smooth.
Place dough in a lightly greased bowl, cover with a damp cloth, and let rise until doubled, about 1 hour.
Preheat oven to 350°F (175°C). Line a baking dish with parchment paper.
Punch down dough and roll out on a floured surface into a rectangle.
Spread softened butter over the dough, sprinkle with brown sugar and cinnamon.
Roll up the dough tightly and cut into even slices. Place rolls in prepared baking dish.
Cover and let rise again for about 30 minutes.
Bake in preheated oven for 25 minutes, or until golden brown.
While rolls are baking, prepare frosting by beating cream cheese, butter, powdered sugar, and vanilla extract until smooth.
Once rolls are baked, spread cream cheese frosting over warm rolls before serving.