Combine chicken stock, potatoes, zucchini, onions, carrots, dill, basil, garlic, thyme, and rosemary in a large pot; bring to a boil. Reduce heat to medium-low and cook soup at a simmer until the vegetables are completely tender, about 40 minutes.
Leaving pot on stove, switch heating element to off.
Add Cheddar cheese to soup; stir until the cheese is melted. Season soup with salt and pepper.