[Anton]

 I thinned out my New Zealand spinach the other day and eat the thinnings. Very tasty. The rest are now growing fast and furious. How doI harvest the others? Do I take off the branches or just the leaves?

Anton

[New shoot]

You can do either.  New Zealand spinach is a fast growing, spreading plant, so if you want it to increase in size, just pick the leaves. 

If it is already outgrowing its space, pick the branches.  You might choose not to eat the stems if they are a bit more mature, but it is a quick way to get a load of leaves into the kitchen