Hard boil the eggs for 8-10 minutes. Peel and dice. Set aside.
In the meantime, heat the avocado oil in a non-stick pan and sauté the diced chicken for 3-5 minutes until crispy and golden. Season with a pinch of salt and pepper.
Add the chopped bacon to the pan and crisp for a few seconds. Set aside.
Next, make the dressing: In a blender or food processor combine the mayonnaise with the olive oil, anchovies, lemon juice, garlic, and Worcestershire sauce. Process until creamy. Season to taste with salt and pepper and optionally adjust the acidity by adding a few extra drops of lemon juice.
Cut wedges out of the iceberg lettuce head and arrange on a serving plate.
Top with the chicken and bacon, cherry tomatoes, and chopped eggs.
Add dollops of the dressing and serve garnished with shaved parmesan.