In the bowl of your stand mixer, mix together the softened or melted butter, milk, heavy cream, egg, and sugar. Ensure the milk and cream are warm to touch (not hot) to create an ideal environment for the sourdough starter.
Add the active sourdough starter to the bowl. Use the paddle attachment on a low speed to ensure it's fully incorporated with the wet ingredients.
Switch to the dough hook attachment. Slowly add in the flour while the mixer is running at a low speed. Continue mixing, scraping down the sides of the bowl as necessary, until a soft dough forms. The dough should start pulling away from the sides but still be slightly tacky to the touch.
Once combined, cover the bowl with a kitchen towel or plastic wrap and let the dough rest for 30 minutes.
After resting, knead the dough with the dough hook on medium speed for about 5 minutes. You're looking for the dough to form a smooth, elastic ball that's not too sticky.
Transfer the dough to a lightly greased bowl, covering it again with a towel or plastic wrap. Allow it to rise at room temperature for 8-12 hours, or until it has visibly grown in size and shows signs of air bubbles.