In a medium saucepan, place the eggs and cover with water by 1-inch. Bring to a boil, reduce the heat to medium-low, and cook the eggs at a bare boil for exactly 10 minutes.
Drain and rinse under cold running water. Place the eggs in an ice bath to cool. When chilled, peel the eggs, cut in half lengthwise, and chop into 1/2-inch pieces. Place in a medium bowl and add 1/4 cup of the mayonnaise, the shallots, garlic, and cayenne and mash with a fork to blend. Add the chopped lobster tail meat and additional mayonnaise 1 tablespoon at a time to reach desired consistency. Adjust the seasoning, to taste. Cover and refrigerate until ready to serve. (Salad can be refrigerated for up to 24 hours.)
In a bowl, toss the mesclun with the olive oil and salt and pepper, to taste. Arrange across the bottom of on a large plate or platter.
With a thin, sharp knife, cut each avocado half lengthwise almost all the way but not completely through to the other end. Place each half seed side down on the lettuce and gently spread out into a fantail shape. Place the lobster-egg salad in the center of the two avocados and the asparagus spears around the sides of the plate. Top the avocados with caviar, garnish with the tarragon, and serve.
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