DOUGH: In the bowl of a stand mixer, sprinkle yeast over warm water. Add teaspoon of sugar and stir to combine. Let sit for 5-7 minutes to allow mixture to bubble and activate.
Whisk in remaining sugar, milk, egg, and egg yolk. Attach dough hook to stand mixer.
In a large bowl, whisk together flour, salt, and cinnamon. Add 2 cups of flour mixture to stand mixing bowl. Mix on low speed until dough starts to come together.
Add another 1 cup of flour mixture followed with shredded carrots. Mix on low speed until carrots are distributed. With the mixing running on low speed, add softened butter one tablespoon at a time. Add remaining ½ cup of flour and knead until the dough is smooth and cleans the sides of the bowl.*
Transfer dough to a lightly greased bowl. Cover with plastic wrap and let dough rest at room temperature for about an hour, until it has doubled in volume.
Punch down risen dough. Transfer to a lightly floured work surface. Roll dough into a 20 x 14 inch rectangle.
Filling: Spread softened butter over rolled out dough. In a small bowl, whisk together sugar, brown sugar, cinnamon, and salt. Distribute cinnamon sugar filling over buttered dough.
Assembly: Starting at the longer end, roll dough into a log. Slice log into 12 equal parts (or 24 equal parts). Place rolls in a greased 9x12-inch baking pan or 9-inch round cast iron skillet. Cover with plastic wrap and let rolls rest for 30-45 minutes at room temperature to allow rolls to rise.
Preheat oven to 350° F. Remove plastic wrap. Bake rolls for 30-35 minutes until golden brown on top and the internal temperature of rolls registers at least 185 degrees F. Allow rolls to rest for 5 minutes to cool slightly.
Frosting: Beat together cream cheese and butter Add vanilla, powdered sugar, brown sugar, and salt. Beat until smooth. Spread over warm carrot cake cinnamon rolls. Top with toasted walnuts, as desired. Serve cinnamon rolls warm.