Melt butter in a large pot over medium heat. Then add in the leeks and allow them to soften for about 5 minutes. Stir so they do not burn.
Add garlic, carrots, and celery, and cook for another 2 minutes.
Pour in the chicken broth and give a few good stirs.
Add the chicken, diced potato, salt, pepper, thyme and dill. Bring to a low simmer and cover for 25-30 minutes. The chicken and potatoes should be cooked through.
Remove the chicken from the pot and shred it with two forks.
Using an immersion blender, blend the soup until it is about half smooth. Leave some smaller pieces of vegetables. Add the chicken back into the pot. (Skip this step if you like a chunky soup)
Add in the lemon juice and stir to combine.
Stir in the heavy cream off heat until even. Bring back to a gentle simmer for 1-2 minutes. Ladle into bowls and enjoy.