Melt butter in a large stockpot or saucepan over medium heat. Add onion, carrots and celery. Cook, stirring occasionally, until tender, about 10 minutes. Add chicken and stir. Continue cooking, stirring occasionally, until the chicken is starting to brown slightly. Stir in thyme and rosemary.
Stir in flour until lightly browned, about 1 minute. Gradually stir in chicken stock, milk, and half & half. Stir until completed combined. Stir in potatoes.
Bring to a boil; reduce heat and simmer until potatoes are tender, about 12-15 minutes. Stir in cheese, a handful at a time, until smooth, about 1-2 minutes; season with salt and pepper, to taste.
Serve immediately, garnished with parsley, if desired.