There are volatile sulfur compound inside the cells of the onion. When those sulfur compounds hits your eyes, it causes them to tear. Note that neither plugging your nose or breathing through your mouth will help. As soon as that sulfur compound hits, your eyes, you'll tear. It has nothing to do with breathing.
This means that any solution will have to keep that sulfur compound fumes away from your eyes will help.
- Blower to blow the fumes away from you.
- Cut the onion underwater, so that the sulfur compounds are dissolved in the water, and don't get into the air.
- Wear onion goggles.
- Refrigerate the onion. The sulfur compounds are less volatile when cold, so they're less likely to get into the air and into your eyes.
- Finally: Use a very sharp non-serated blade knife to cut the onion.
Peel the onion by making a cut through the outer layers (not too deep) and try to remove the peel without tearing it. If you are a bit messy, that's okay, wash the onion off under cold running water as soon as the peel is removed to get rid of the liquid that contains the sulfur compounds.
Now, chop the onion. Make sure the knife is very sharp. Sharpen the knife before you cut it. Again, do not use a serrated knife. If you can't sharpen a non-serated knife sharp enough to cleanly cut through a soft tomato, it's not sharp enough. If you can't seem to get the knife sharp enough with a sharpening stone, get something like the Chef's Choice Knife Sharpener. It's not a character flaw if you can't sharpen a knife on a sharpening stone, so don't feel ashamed. I in fact, have that very model of knife sharpener.
Cut the onion with the knife. Slice through it. You want to make sure you don't tear the cells of the onion. Don't hack the onion to death. You're not a serial killer. You want clean cuts. Thin slices, and then cut through the slices perpendicularly.
You should now have fairly small onion slices, and this may be enough for most cooking tasks. If you need a finer consistency, you can chop the onion.
If you watch this video, you'll see how this chef slices through the onion and never hacks at it -- even when he is making cross hatches in the onion. Notice he sharpens the knifes before cutting the onion. Notice the knife he's using -- a big sharp and very fine bladed chef knife.
Okay, he doesn't rinse the onion after peeling, but he probably chops it fast enough that it's not a problem. Besides, he's a professional, so don't try that at home - rinse the onion after peeling.