So I went to Subway™ and saw that their carrots are cut into thin strips. I can never do this with any knives. So what is the best way to cut carrots into thin strips (like french fries)?
The style of cut is called julienne. True, a mandoline can make a julienne cut, as well as thin slices. However, you can also use a knife to achieve a julienne cut.
First, cut the carrot into manageable lengths--2 or 3 inches. Cut a piece into 1/8 inch slices lengthwise. Stack several slices on top of one another and cut lengthwise through all layers to make "matchsticks" or julienne-cut carrots. It works best to have a large slice on the bottom of your stack, not a small one or the round side of the carrot. A large slice on the bottom is more stable, so you're not as likely to julienne your fingers. Yes, cutting carrots this way is a lot of trouble, but this is the way to do it with a knife.
There are also other ways to get a similar result without a mandoline. If you have a food processor, try the large grating disk. You can also use a salad shooter or other slicer/shredder--there's even an attachment for the ubiquitous Kitchen Aid mixer. Even a plain old box grater can be used to grate carrots. Grated carrots may not be as neat and tidy as the ones done commercially, but they're still a good addition to a salad.
Oh, if all else fails, you can usually buy the shredded carrots in the produce section, near the bagged salads.
If you're talking about fast food, you can rest assured that their carrots arrive that way from the distributor, who is probably using industrial machinery to do the cutting.
If you want to achieve this at home with minimal effort then your best bet is a piece of equipment called a mandoline. Normally it has a top piece that you use to pierce the vegetable or fruit (AKA a guard), and you just slide it along the surface as it gets sliced by the slightly-raised blade, which you can usually adjust to get your desired thickness. It looks like this:
Most of them have julienne blades but make sure you check before purchasing one.
Safety note: As contributor Chef has pointed out, you can give yourself an extremely nasty cut on one of these if you get careless (and who among us doesn't have those days occasionally?), so do yourself a favour and buy a pair of safety gloves if you don't already have them.
Gloves are not expensive - the most expensive pair on Amazon costs under $25 - so just go and get yourself a pair, even if you have a really high-end mandoline. It's better to have a good pair and never use them than it is to slice half a finger off because you couldn't spare the time or expense to get some. Trust me, I'm speaking from experience here.
Subway uses specialized electric appliances to do such cuts. The closest thing at home is the food processor, but that doesn't cut that thin. In resraurants, we get things thin by using a mandoline, as others have pointed out. What they haven't pointed out is how dangerous thy are. Thy are probably the single largest contributor to cuts- nasty ones too. It is VERY easy to slip, or have the product slip ( soft/old carrot?) and cut your fingertip. Cooks really should use a guard- they never do. I wouldn't recommend using a mandolin for most home cooks- you will cut yourself eventually- and badly. If you DO use one, make sure you use a decent guard. The plastic piece of junk that comes with the benriner mandolin isn't very good. Another alternative use a peeler, not as nice, but it can work.