The way I am reading your question. Compost the tops of mallow and leave the roots in the soil.

Answer maybe, maybe not. Depends on the age of the mallow and it's type. If you can use a hoe and cut the tops off below the crown yes. If you just cut the top foliage off no. There is no problem composting mallow but by leaving the crown it will send new growth up from it. Better to cut off below crown.

Which of the Mallows are you refering to?

I was told it's called "cheese" mallow?

Something like this?

Here is a link that might be useful: Cheese Mallow