Activate the yeast. Warm the milk to about 110°F. Add the 1 tablespoon of sugar and the active dry yeast, then stir. Let sit for 5–10 minutes, until foamy.
Mix the dough. In the bowl of a stand mixer, combine the activated yeast mixture, egg, ¼ cup sugar, 3 cups of flour, and salt. Using the dough hook, mix on medium-low speed for 8–10 minutes.
Add the butter. With the mixer running on low, add the softened butter 1 tablespoon at a time until fully incorporated. Add the remaining ¼ cup of flour and continue kneading for another 8–10 minutes. The dough should be smooth, slightly tacky. Transfer to a lightly greased bowl and cover.
Proof. Let the dough rise in a warm spot until almost doubled in size, about 1–1 ½ hours.
Make the skillet caramel. As soon as the dough begins proofing, prepare the caramel base. In a 10-inch cast-iron skillet, combine the butter, brown sugar, and heavy cream. Cook over medium heat, stirring occasionally until melted and beginning to simmer. Let it simmer undisturbed for 1 minute, then remove from heat. Allow the caramel to cool completely in the skillet. (It must be fully cooled before assembling the cinnamon roll—warm caramel will cause the dough to overproof during the final rise.)
Make the filling. In a small bowl, mix together the softened butter, brown sugar, cinnamon, honey, and salt.
Roll out the dough. Once the dough has nearly doubled, punch it down and turn it out onto a lightly floured surface. Roll into a 12 × 15-inch rectangle.
Add the filling. Spread the filling evenly over the entire surface of the dough.
Cut and assemble. Using a pizza cutter, cut the dough horizontally into 6 long strips (about 2 inches wide).Starting with one strip, roll it up. Place that roll at the end of the next strip and continue rolling. Repeat one more time, then transfer the partially formed roll into the skillet. Wrap the remaining strips around it, slightly overlapping the end of each strip with the beginning of the next. Final proof. Cover the skillet with plastic wrap or a tea towel and let the cinnamon roll proof in a warm spot until puffy and slow to spring back when gently pressed, about 1 hour.Overnight option: Cover and refrigerate overnight. In the morning, let it sit at room temperature until the dough springs back slowly, about 1 ½–2 hours (depending on kitchen temperature). Preheat and bake. When the roll has been proofing for about 30 minutes, preheat the oven to 350°F.When the dough is ready and fully proofed, place the skillet on a baking sheet (to catch any caramel drips) and bake for 30–35 minutes, until the center registers about 190°F on a thermometer. Check around the 20-minute mark; if it’s browning too quickly, loosely tent with foil. Make the icing. In a small bowl, whisk together the softened butter, heavy cream, powdered sugar, salt, and vanilla until smooth.
Ice and enjoy! Let the cinnamon roll cool for about 5 minutes, then spread the icing over the top. Slice and serve—or pull apart straight from the skillet!