Follow the steps in order for the best result.
Heat the cooking oil in a large, heavy-bottomed skillet with a lid over medium-high heat. Add the diced onion, sliced sausage, and sliced potatoes to the skillet as you prepare them.
In the skillet, incorporate the brown sugar, applesauce, mustard, cider vinegar, thyme, and black pepper according to your preference.
Stir all the ingredients together, then cover the skillet. Continue to stir approximately every 5 minutes, allowing the potatoes to become tender, which should take around 25 minutes.
Remove the lid from the skillet. Continue to cook and stir at 5-minute intervals until the mixture achieves your desired level of browning.