In the bowl of a stand mixer or a large mixing bowl, add the warm milk and yeast. Stir to combine and let sit until frothy, about 5 minutes.
Add the brown sugar, shredded carrot, kosher salt, soft butter, and egg. Stir briefly, then add the flour.
Using the dough hook attachment, knead on medium speed for 7–10 minutes. If mixing by hand, stir until the dough comes together, then knead on a lightly floured surface for 7–10 minutes until smooth and elastic.
Cover the bowl lightly with a kitchen towel and let the dough rise in a warm place for 1 hour 30 minutes.
Preheat your oven to 350°F.
Prepare your 9x13-inch pan by spreading the soft butter across the bottom. Sprinkle the brown sugar evenly, then drizzle the heavy cream on top.
Roll the risen dough into a large rectangle about 10x14 inches on a lightly floured surface.
Spread the melted butter evenly over the surface of the dough. Sprinkle with brown sugar, followed by cinnamon and salt.
Roll the dough tightly, starting from the edge farthest from you, rolling toward you. Finish with the seam side down.
Slice into 6–8 even rolls using a sharp knife or bench scraper.
Place the rolls into the prepared pan. Let them rise again for 30 minutes.
Bake for 35–40 minutes. Start checking at 30 minutes, as oven temperatures may vary. The rolls should be golden brown and bubbling.
While the rolls bake, make the icing: whip the cream cheese for 2–3 minutes until smooth. Slowly add the powdered sugar, then pour in the heavy cream while mixing. Add a pinch of salt. Beat until light and fluffy.
Once the rolls are baked, let them cool for 5–10 minutes before spreading the icing generously over the warm rolls. Enjoy every cozy bite.