Put a stock pot on medium high heat. Add the stock.
Season the chicken with ½ teaspoon kosher salt and fresh pepper. Add the chicken to the stock.
Bring the stock up to a boil then back to a simmer. Skim the stock as needed if a white foam forms. Begin checking the chicken for doneness at 8 minutes. You will want the internal temperature to read 165°F.
Once the chicken is done, remove from the stock and shred with two forks. Keep the stock at a low simmer. Set the chicken aside.
In the heavy-bottomed pan, add the olive oil. Set to medium low heat. Add the leeks and a pinch of your remaining salt. Cook for approximately 10 minutes, until the leeks are tender. Be sure to stir often so the leeks do not brown.
Add 1 cup of the simmering stock to the leeks and let evaporate, stirring occasionally. Turn heat to medium high.
Add the rice. Toast the rice until the exterior is translucent but the interior remains white, stirring consistently. Continue to stir and watch heat so as not to burn the leeks.
Add the garlic and stir for 30 seconds. Add the vermouth or white wine. Let cook until the liquid evaporates.
Slowly add the broth to the rice 1 cup at a time. Stir the broth in the rice until the liquid has fully evaporated. Then add the next cup.
Continue this process until the risotto reaches an al dente texture, or cook for a bit longer if you prefer a softer risotto. Start tasting for doneness around 15 minutes.
Test for consistency. You don’t want it soupy but you do want it loose enough that it spreads a bit when you put it on a plate and shake the plate back and forth on the counter. You can test it with a scoop before your final plating.
Once the risotto is cooked, taste for seasoning and add more of remaining salt if needed. Stir in the shredded chicken and the fresh dill.
Serve immediately with some fresh squeezed lemon juice and shaved parmesan.