Bring a pot of water to a boil and blanch the grated carrots and minced celery for 30 seconds. Scoop out the vegetables and put them in a bowl of ice cold water to stop them from cooking.
Melt the butter over low heat. Add the onion, thyme, and bay leaf, raise the heat a bit and cook, stirring, about 5 minutes.
Lower the heat and stir in the flour and cook about 2 minutes, stirring most of the time. Add the hot broth a cup or so at a time and whisk after each addition until the soup is smooth.
Stir in half the Cheddar until melted, then stir in the other half until melted.
Drain the carrots and celery and add to the soup along with the half and half. Taste and add salt and pepper. Garnish with the fresh parsley.