My goal is to shred carrots to use in a green salad. I've used graters with differently-sized holes, but it seems whenever I shred carrots, the shredded bits are very mushy and clump together. How do I get the nice, dry shredded results like I find in restaurants or in the salad packs at the grocery store?
The best method I've found for this is to use a vegetable peeler. This results in nice, beautiful ribbons with no mush at all.
I recently used this method when making homemade cole slaw and it gave a wonderful texture and presentation. I'm not sure if it's the size you're looking for, but given how well it worked with the cole slaw I made, I can't imagine it couldn't be used successfully for a salad.
To do this, you'd simply peel the carrot, discard any dirty peels, and then continue to "peel" the carrot. The strips you get will be even, clean, and without any mush at all.
If this method is not to your liking, certain food processors have attachments which may be able to give you results similar to the shredded carrot you can find pre-packaged in the store. Yet, I couldn't say if this would result in a mushy consistency, as I haven't used that method myself.
Update: Note my comment below. There is a device known as a "carrot shredder" which may be what you're looking for. It looks like a vegetable peeler, but has notches on it. This results in shreds instead of ribbons.
One of the issues, I think, is that a duller blade will bruise the carrot more - pull a little more liquid out by tearing the carrot, rather than slicing cleanly. Box graters depend on the force and speed of the hand grating, rather than the sharpness of the metal - you might find a sharp mandolin slicer better for the kind of shredded carrot you want. An inexpensive mandolin slicer might be more effective than the box cutter, but may be less so than a better one - it depends on how sharp it is.
Other possible factors might include the size of the shreddings (width and thickness both) since smaller shreds will have more surface area and opportunity for the carrot juice to present itself, or the moisture level of the carrots themselves - which may be linked to freshness or temperature or all sorts of things. Drier, firmer carrots will probably not clump as badly, fresher carrots (with more snap and less bend) might slice more cleanly and so leak less moisture. Grated carrots (torn little uneven fragments) will be wetter and more mushy than shredded carrots (which are longer and more thread-like).
And , finally, it is also possible that the commercial shredded carrots are somehow treated before being sold in the store or used in a restaurant - being stored in shredded form might allow them to air-dry a bit, or they are let sit until extra juice drains to the bottom of their container and then fluffed for presentation, or extra moisture is blotted away with a towel or something, or they're dusted with a fine layer of starch (like shredded cheese) especially to minimize clumping.