Combine chicken stock, sweet potatoes, ham, celery, and onion in a stockpot; bring to a boil. Reduce heat to medium and cook until sweet potatoes are tender, 10 to 15 minutes. Stir green bell pepper, carrot, corn, garlic, basil, parsley flakes, black pepper, and salt into soup and simmer until vegetables are softened, 5 to 10 minutes.
Melt butter in a saucepan over medium-low heat; whisk flour into butter using a fork. Cook and stir mixture until thickened, about 1 minute. Slowly stir milk into butter-flour mixture until completely smooth; continue cooking and stirring over medium-low heat until thickened, 4 to 5 more minutes. Stir Cheddar cheese into milk mixture until melted and smooth.
Stir milk-Cheddar cheese mixture into soup; cook until thickened and heated through, about 5 minutes.
