Preheat the oven to 325 F
Add the chicken to a dutch oven with 1/2 an onion, 1 celery stick, 1 carrot, and some of the green leek tops. No need to chop the veg, we're just making the broth here and will throw them out afterwords.
Add about 5 quarts of water. Enough to cover the chicken and veg by about an inch or two. Add the salt.
Put the lid on the pot and place in the oven for 2.5-3 hours. When the chicken falls easily off the bone, it's done.
Prep the remaining vegetables: dice the onion and white part of the leek; slice the carrot and celery; and mince the garlic.
Remove the pieces of meat and place on a cutting board to shred the chicken. Put the shredded chicken meat in a bowl and discard the bones. Discard the cooked vegetables.
Strain the stock you just made through a metal sieve into a large bowl with a pour spout. Set aside.
Place the dutch oven back on the burner at medium heat. Add 3tbsp of butter or ghee to the pot. Once it's hot, add the chopped veg and season with salt. Saute for 10-15 minutes until they begin to soften but not brown. Add the garlic along with another 3-4tbsp of butter and stir for one minute. Add the broth back to the pot along with the thyme and/or bay leaf. Bring it up to a simmer for 5 minutes then add the shredded chicken to the soup. Let it simmer lightly for 5 more minutes. Taste for seasonings. Add more salt if needed, add some pepper.
Top with a squeeze of fresh lemon juice, some freshly chopped dill, parmesan cheese if desired, and a heaping spoon or four of sour cream.