Make the sauce first. In a bowl, combine all the ingredients except the olive oil and green onions. Add the oil in a thin stream while whisking. Stir in the green onions. Refrigerate.
Put the eggs into a saucepan and cover with an inch of cold water. Turn on the heat to medium. When the water comes to a boil, turn off the heat and remove the pan from the burger. Cover the pan and let the eggs sit in the hot water for 12-17 minutes, depending on the size of the eggs. (the bigger, the longer.) Remove the eggs and allow to cool.
Peel the eggs by first cracking the shell on the counter, in a few places. Then roll the egg on the counter while pressing down a bit. Begin peeling from the end with the depression in the white. Try to peel it in a spiral. (You won't be able to, but that motion will get the peel off most effectively.)
Slice the eggs in half lengthwise. Scoop out the yolks into a bowl. Set the whites aside.
Combine the yolks with the onions, celery, capers, mayonnaise, mustard, salt, and Tabasco. Mix well with a whisk (you can even beat it to fluffiness).
Load the mixture into a pastry bag and pipe it into the centers of the boiled egg whites.
Slice the avocados in half and remove the pit and skin. Slice the avocado about as thick as a quarter, but retain its shape. Fan out the slices on a plate and put the egg where the pit was.
Drizzle the remoulade sauce generously over everything and serve chilled. For an extra touch, a couple of boiled, peeled shrimp could be added.