Melt butter in a pot over medium-high heat. Add onions and garlic; saute until softened, about 6 minutes. Add squash, chicken broth, and marjoram. Reduce heat to low. Simmer soup until squash begins to soften, about 20 minutes. Stir in cream cheese until melted.
Pour soup into a blender and puree until smooth. Pour back into the pot.
Stir Cheddar cheese blend, pepper Jack cheese, and cornstarch together in a large bowl until cheese is coated. Mix into the soup and simmer until well blended, about 5 minutes. Add half-and-half; mix until smooth.
