You can do both and yes, it will continue to grow (slowly) if you cut off individual leaves.

Say, Codywalker,
I've taken Eliot Coleman's advice, and I grow mache in my coldframes, along with some other all-winter vegetables he suggests in his book, Four Season Harvest. One tip: I discovered that it is best to havest mache with a small sharp knife, as he suggests. I had reasoned that cutting it with scissors was the superior way. But the sharp knife simply works better. ---Bunn

I don't know my zone but I live in Western Ky. This is my first year planting Mache (Corn Salad. I never heard of it until this year. Can I plant it now in April 9th...?