In a small bowl, combine tarragon, salt, pepper, and olive oil. Spread over chicken breasts to season.
I cooked the chicken in our George Foreman Grill for about 7 minutes. You could also prepare the chicken in a skillet on medium-high until fully cooked.
Separate eggs, putting yolks in a small bowl and the whites into a medium mixing bowl. Beat yolks.
Beat whites until light and fluffy. Set both aside.
With two forks, shred the chicken breasts and put into a bowl; set aside.
Pour chicken broth into a large saucepan and bring to a boil over medium-high heat.
Add orzo and cook for two minutes less than the required cooking time.
Add chicken and pepper to the pan and cook contents for another two minutes.
Reduce heat to medium-low.
With a whisk, begin whisking the egg whites as you slowly pour the egg yolks in.
Continue whisking as you slowly add the lemon juice, emulsifying the mixture.
Add about 1 cup of just the broth from the heated pan to the eggs.
Add the lemon/egg mixture to the soup, and add pepper to taste.
Gently shake the pot instead of wanting to stir, as this was suggested by a true Greek.
Heat over medium-low heat for 2-3 minutes without covering.