In a Dutch oven or large soup pot brown the chicken breaking into small pieces as it cooks. Brown the chicken until there is no pink showing.
Once the chicken is cooked remove it from the Dutch oven and set aside.
Melt the 1/4 cup butter in the Dutch oven. Add the onions, potatoes, carrots, and celery. Cook for 8 to 10 minutes or until the vegetables start to get tender. Stir often so that the vegetables do not burn.
Stir in the flour until it is blended into the vegetables. Slowly add the half and half, stirring until blended in. Add the chicken broth and stir in.
Add the bay leaf, bouillon, and cheese, stir until the cheese is melted.
Bring the chowder to a boil and reduce heat and continue to simmer/cook for another 20 minutes or until vegetables are fork-tender. Season with salt and pepper.