There is no need to finely chop here, you'll be adding all ingredients to a blender later, so just chop the onions enough that you can saute, and break the cauliflower apart with your hands into manageable pieces.
Cheese options:
We have made this recipe half a dozen times already, and honestly it tastes pretty darn good with just about any cheese. BUT our favorite combination is ½ cup Gruyere, ½ cup pepper jack, and 2 cups sharp cheddar.
Feel free to just use 3 cups sharp or medium cheddar if you don't have the others on hand...but the pepper jack definitely gives this soup more flavor!
DO NOT use pre-grated cheese. These products are tossed in starch and won't melt as well, and the texture will be off. Grate your own cheese.
Using mild cheddar will result in a soup that might taste bland, we don't recommend it..