Turn Instant Pot to saute mode and heat 2 tbsp olive oil. Add carrot, onions, and celery. Saute until softened, about 6-7 minutes. Add garlic and saute for another minute. Turn off Instant Pot.
Add chicken, chicken broth, lemon juice, lemon zest, salt, pepper, and red pepper flakes. Cook on high pressure for 10 minutes. Let natural release for at least 15 minutes. When time to release, use a long cooking spoon to open valve in case any soup spits out through valve.
Remove chicken breasts from soup and shred with two spoons. Set aside.
Use an immersion blender to blend soup in pot until desired mostly smooth, but some small chunks remain. Remove 1 cup of soup and set aside.
Turn Instant Pot to saute mode and bring soup to a simmer. Add dry pearl couscous and simmer for 8 minutes. Turn off Instant Pot and remove inner pot. Set aside to cool for 2-3 minutes before next step.
In a small bowl, whisk the 3 eggs. While constantly whisking, very slowly pour in the cup of soup that you had set aside. Then, while constantly whisking soup in inner pot, pour egg/soup mixture in.
Stir in chicken. Serve with feta and green onions.
Note: This soup is very hearty. If you prefer more broth, I recommend adding up to one cup of additional broth at the end.