Place the eggs in a small pot and cover with cool water. Place the pan over medium-high heat, cover, and bring to a boil.
When the pan has come to a full boil, turn off the heat and leave the pan untouched for exactly ten minutes.
When ten minutes has passed, pour off the water and rinse the eggs under cold water to stop the cooking.
Peel the eggs, cut in half, and remove the yolks from the whites. Roughly chop the whites.
In a mixing bowl, mash the yolks with the mayonnaise, salt, and pepper until creamy. Mix the celery and green onion into the yolk mixture, then stir in the chopped whites. Taste and adjust seasoning as desired.
Set aside half of the egg salad to use for your meal with an avocado and store the remaining half in an airtight container in the refrigerator for up to three days.
Season the avocado with salt and pepper, then fill the center with egg salad, and garnish with cucumber slices.