Beat eggs, 2 tablespoons chives and milk in bowl to blend. Melt butter in heavy large skillet over medium heat. Add onion and sauté until golden, about 15 minutes. Add egg mixture. Cook until almost set, stirring occasionally, about 4 minutes. Mix in salmon. Cook until eggs are cooked through but still moist, about 1 minute longer. Season with salt and pepper. Transfer eggs to platter; sprinkle with remaining 2 tablespoons chives.