Make the broth: In a large saucepan, add the chicken thighs, about a quarter of the leek, one carrot, a few pieces of celery, plus salt and pepper. Pour in enough water to cover and simmer until the chicken is cooked through and the veggies are tender. Once done, remove the chicken and set it aside to cool. Strain the broth, but don’t toss those veggies — you’ll need them later.
Sauté the aromatics: In a big soup pot, melt half of the butter over medium heat. Add the remaining leek, carrots, and celery, and sauté for about 5 minutes, until everything smells amazing and starts to soften. Add the garlic and cook for another minute.
Build the base: Toss in the chopped potatoes and your seasonings. Give them a quick stir for a couple of minutes, then pour in your homemade chicken broth along with the whole potatoes. Bring it all to a boil, then lower the heat and simmer for about 15–20 minutes, until the potatoes are nice and tender.
Add the chicken: Shred your cooked chicken and stir it into the soup about 10 minutes before it’s done simmering.
Blend it creamy: Grab the whole potatoes and the veggies (carrot, celery, leek) you saved from the broth, and add them to a bowl with a few spoonfuls of the soup broth. Blend until smooth, then stir that mixture back into the pot — this gives the soup that silky, velvety texture. In a separate bowl, mix the sour cream with a few tablespoons of hot broth, then stir it into the soup as well.
Add the broccoli: Now, toss in the broccoli, cover, and let the soup simmer gently for 5–6 minutes, until the broccoli is tender but still a little bright.
Finish it off: Turn off the heat and stir in the Parmesan and the rest of the butter until melted and creamy. Give it a taste and adjust the seasoning with more salt, pepper, or even a little lemon juice for brightness.
Serve: Ladle it up, top with a sprinkle of fresh parsley or thyme, and enjoy a cozy, comforting bowl of chicken broccoli goodness.