In a medium bowl, mix the cheese mixture, heavy cream, thyme, salt, and pepper. Heat the butter on your pan or skillet to coat it (use more if necessary). Place the cheesy cream mixture into the pan and cook for 1-2 minutes over medium until the mixture starts to bubble.
Then, crack 4 eggs on top of the mixture and season with black pepper. You know it's done with the egg whites start to whiten up, which will ensure the yolk is runny.
If you're using a pan, after the eggs whites form, place the heat on low and cover the pan for another 3 minutes. Skillet cook the cream mixture faster, where as pans take a little more TLC. Season with a pinch more salt and serve it up with mimosas!