In a small saucepan or microwave-safe bowl, heat the milk and butter just until warm to the touch, about 100°F (38°C). If you’re using the microwave, heat it in short bursts so it doesn’t get too hot. You want it warm, not hot. Pour the warm milk and butter into a large mixing bowl or the bowl of a stand mixer. Whisk in the active dry yeast and 1 tablespoon of the sugar. Let it sit for about 5 minutes, until the yeast is bubbly and frothy.
Whisk in the remaining sugar, the sourdough discard, and the egg until well combined. Add in the flour and salt.
Use the dough hook attachment on your stand mixer to knead the dough for 5 - 8 minutes on low speed. If kneading by hand, turn the dough out onto a lightly floured surface and knead for 8 - 10 minutes until smooth and stretchy. If the dough feels very dry, add 1 teaspoon of milk at a time. If it’s too wet, sprinkle in a little flour. You’re aiming for a soft, tacky dough, not dry.Once kneaded, place the dough into a lightly greased bowl and cover it with a towel or plastic wrap. Let it rest for 10 - 15 minutes while you make the filling. In a small bowl, stir together the brown sugar and cinnamon.After resting, roll the dough out into a 16 × 12 inch rectangle. Spread the extra-softened butter evenly over the surface. Then sprinkle the cinnamon sugar mixture all the way across, edge to edge! Starting on the long side, roll the dough up into a tight log. Try to keep it even as you go. Use a sharp knife or unflavored dental floss to slice the log into 12 rolls. Place the rolls in a greased 9 × 13-inch baking dish, spaced just slightly apart to allow room for the rise.
Cover the dish with a clean kitchen towel or plastic wrap and let the rolls rise in a warm spot for 60 - 90 minutes, or until noticeably puffy and doubled in size.
Preheat the oven to 375°F (190°C). Bake the rolls for 22 - 25 minutes until the tops are lightly golden. If they’re browning too quickly, loosely tent the pan with foil partway through. To be sure they’re baked through, check that the internal temperature is around 195 - 200°F (91 - 93°C).
While the rolls bake (or cool), whisk together the powdered sugar, melted butter, milk, and vanilla extract until smooth. Add more milk as needed for your preferred consistency, spreadable or pourable.
Once the rolls are out of the oven, let them cool slightly, then spread the icing over them while still warm so it melts into all the swirls. Serve warm and enjoy every gooey bite!