Finely chop half the olives; set aside. Thinly slice the remaining olives; set aside.
Remove yolks from eggs; set aside. Place whites in a bowl. Using a fork, mash egg whites until 1/4-inch pieces are formed. Add mayonnaise, tuna, reserved chopped olives, capers, and chopped tarragon; stir to combine. Season with salt and pepper. Place 1 tablespoon mixture onto each crostini.
Place reserved yolks in a sieve. With a spoon, press yolks through sieve into a bowl. Top each crostini with yolk, garnish with tarragon and reserved olives, and serve.