First, hard boil your egg. Place egg in a small pot. Cover with cold water. Bring water to a roiling boil, cover, and remove from heat. Let stand in hot water for 10 minutes, then remove egg and run it under cold water to stop the cooking. Peel and chop egg.
Next, melt 1 tablespoon butter in a small skillet. Add onions and cook until soft. Add in chicken liver and cook for about 5 minutes, flipping occasionally. Liver should be nicely browned and easily broken down with a fork. Let cool.
In a small food processor, combine liver and onions, hard boiled egg, red wine vinegar, thyme, remaining 1½ tablespoons butter, salt, and cayenne. Pulse until smooth. Adjust seasoning to taste. Transfer pâté to a small jar and let cool for at least an hour in the fridge, until butter has time to firm up again.