Prep the chicken. Remove the chicken from its packaging and remove any giblets or the neck from the cavity and discard. Pat the chicken dry, inside and out, with paper towels and place it on a small baking sheet lined with parchment paper. Season the chicken all over and in the cavity with salt and pepper. Let the chicken sit uncovered in the fridge for 1 hour.
Preheat an oven to 325 F. Arrange your oven rack so that it sits in the lower third of the oven.
Sear the chicken. Heat the olive oil in a large (7 qt) oval dutch oven over medium heat. Once the oil is hot and shimmering, carefully lower the chicken into the pot, breast side down. Sear for 5-6 minutes, until the skin is deeply browned, then carefully flip the chicken over and sear for another 4-5 minutes. I find the easiest way to flip the chicken is to use a pair of long tongs in the cavity which gives the best control. Once the chicken is browned, transfer it to a plate to rest.
Saute the veggies. Add the butter to the dutch oven and once it’s melted, add the onion, leeks, fennel, garlic, and carrots. Saute for 3-4 minutes, until the vegetables are starting to soften. As the veggies cook, use any liquid they release to help scrape up any browned bits on the bottom of the pan.
Build the soup. Scoot the veggies to the edge of the dutch oven to create a well in the middle. Return the chicken to the pot, breast side up, and nestle it into the veggies. Pour in the chicken stock and then add just enough water (if needed!) to bring the liquid level up so that only a bit of the breasts are above the water line. Stir in the saffron threads, aleppo, and bay leaf and then generously season the broth with salt and pepper. Bring the liquid to a boil, then nestle in the sprigs of dill, parsley, and the parmesan rind (if using), cover, and transfer to the oven.
Braise. Braise the chicken for 75 minutes, then remove it from the oven. Discard the bay leaf, parmesan rind, dill, and parsley sprigs, then stir in the orzo. Cover and return the pot to the oven for 15 minutes more. Remove it from the oven and let it rest for 5-10 minutes. Squeeze the lemon juice all over the chicken and in the broth and season to taste as needed with salt and pepper.
Make the gremolata. While the chicken braises, combine the chopped dill, parsley, lemon zest, grated garlic, and olive oil in a small bowl and stir everything together. Season with salt and pepper and set aside.
Serve. To serve, ladle some of the broth, veggies, and orzo into a bowl. Use tongs to pull some of the meat off the chicken and add it to your bowl. Add dollops of the gremolata and finish with a mound of freshly grated parmesan. Serve right away.