In a large pot or Dutch oven, add the olive over medium heat. Add the diced onion, potatoes, carrots, and celery to the pan. Sauté for about 5-6 minutes, stirring occasionally, until the vegetables start to soften and the onion becomes translucent. Add the minced garlic and stir for about 30 seconds until it smells fragrant. Don't let it burn.
Slowly pour in the chicken broth while stirring continuously. Let the soup simmer for 15-20 minutes, stirring occasionally, until the potatoes are fork-tender. You should be able to easily pierce them with a fork.
Stir in the shredded chicken and half and half. Let everything heat through for about 3-4 minutes. Don't let it boil after adding the cream or it might curdle.
Turn the heat to low. Add the shredded cheddar cheese gradually, stirring after each addition until it melts completely into the soup. The soup should be thick and creamy now. Taste the soup and add salt and pepper as needed. Stir in the fresh parsley and top with more cheese right before serving.