Warm Instant Pot over medium heat. Add olive oil and saute carrots, celery, leeks and garlic for 6-8 minutes, stirring occasionally.
Add chicken broth, oats or rice (whichever using), poultry seasoning, salt and pepper and stir to combine.
Lock the lid and set Instant Pot to PRESSURE, HIGH for 6 minutes. Once cooking time is over, let the pot sit for 10 minutes; then shift the pressure valve to VENTING to release remaining steam. (If cooking this on the stove top, bring to a gentle simmer, cover and cook on a low heat, stirring every 5-10 minutes, until the oats/rice (whichever using) are cooked but slightly al dente (this should take about 20-25 minutes).
Stir the shredded chicken into the soup and cook on SAUTE, MEDIUM (stovetop: medium heat) for another 5 minutes.
While the soup is simmering, whisk the eggs and 2 tablespoons of the lemon juice in a small bowl until thoroughly combined. To temper the lemon-egg mixture, slowly pour a ladleful of the hot soup into the mixture, whisking it continuously as you do so. Then pour the lemon-egg mixture back into the hot soup and stir to combine. The soup will turn creamy and thicken further. Turn off the Instant Pot (stove top: remove from heat).
Season to taste with salt, pepper, additional lemon juice if desired, and a dash of Thai fish sauce (optional but highly recommended; it won't make the soup taste of fish, it will simply deepen all the flavors of the soup in the yummiest way imaginable...).
Stir in the chopped dill and/or parsley and serve, garnishing each bowl with a sprig of dill or parsley. Serve with fresh lemon wedges.