Preheat the oven to 350 degrees.
In a large skillet (or cast iron pan) over medium heat, cook the cudighi sausage. Remove sausage from the pan, leaving the drippings.
Microwave the potatoes to give them a head start on the cooking process. Place the potatoes in a shallow bowl and microwave for 4 minutes, covered.
Add the vegetable oil to the pan. Once the oil is heated, add the potatoes, salt, and fresh ground pepper. Let the potatoes cook for 15-20 minutes, turning over every 5 minutes (stirring too often will not allow the potatoes to brown). After 10 minutes add the onions.
Reduce the heat to low. Add the sausage and curds to the pan.
Create four shallow wells in the sausage hash. Crack an egg into each well. Cover and cook for 8-10 minutes, or until the eggs are set. If you like a firmer egg, cook up to an additional 2-3 minutes.
Top with fresh parsley and serve with dill sauce on the side for dipping.