Place your eggs in a single layer on the bottom of your pot and cover with cold water. The water should be about an inch or so higher than the eggs. Cover the pot with a lid. Over high heat, bring your eggs to a rolling boil. Remove from heat and let stand in water for 12 minutes for large eggs. Reduce the time slightly for smaller eggs, and increase the standing time for extra-large eggs. Drain water and immediately run cold water over eggs until cooled.
Place eggs in a large ice water-filled bowl. While eggs cook, combine 3 tablespoons water, vinegar, and sugar in a medium microwave-safe bowl; microwave at HIGH 2 minutes or until boiling. Add celery; let stand 15 minutes. Drain.
Meanwhile, combine avocado, mayonnaise, lemon juice, mustard, pepper, and salt in a medium bowl, stirring well until smooth.
Peel eggs; discard shells. Slice eggs in half lengthwise; reserve 2 yolks for another use. Chop remaining eggs and egg whites. Gently stir eggs, celery, and sunflower seeds into avocado mixture. Top 4 bread slices with about 1/2 cup egg mixture, 1/4 cup arugula, 1 tomato slice, 2 pieces of turkey bacon, and remaining 4 bread slices.