In a large sauté pan or Dutch oven, heat 2 tablespoons olive oil over medium heat. Add onion slices and 1 teaspoon salt.
Sauté for about 10 minutes until onions are soft and starting to brown, stirring often, and then turn down the heat to medium low and sauté until onions are golden brown and caramelized, about 45 minutes. Stir often. You want the onions to stick a little so they color, but if onions start to burn, add a little water or more olive oil (or wine!) to deglaze. Add 1 more tablespoon olive oil and mushrooms and sauté for 10 more minutes until soft but not mushy. Let cool.
Put onions mixture, eggs, and walnuts, in a food processor and pulse until ingredients are combined and uniform and it looks like chopped liver; add a little olive oil if it's not coming together. Season with salt and pepper to taste.
Chill for one hour or until ready to serve and garnish with walnuts and chives. Serve with matzah crackers and veggies.