Preheat oven to 325°F (162°C).Add oil to medium skillet and heat over medium heat. When oil begins to smoke, add chopped onion and cook, stirring occasionally until softened and they have turned a light amber color (about 20 minutes).Remove from heat and set aside.
Line the bottom of 6 8-oz ramekins with Swiss cheese (2 slices per ramekin). Top each with about 1/4 cup of onion. Crack one egg into each ramekin and sprinkle with a little salt and pepper. Set aside.
In a medium bowl, add cream and wine. Beat with a wire whisk until well blended (about 1 minute). Pour mixture around eggs, dividing evenly between ramekins.
Place ramekins on a cookie sheet and bake uncovered until eggs are set to your liking using the guide below:runny yolks: 23-25 minutesset yolks: 35-40 minutesRemove from the oven and allow to cool slightly. Garnish with chopped chives. Serve with buttered sourdough toast for dipping or you can use the egg mixture as a spread.