I did this SAME thing a couple weeks ago! Same finger, same side of the nail, same tool...although I was cutting potatoes.
I know it is too late, but this is what I did:
I used a butterfly closure to pull the skin together as best as I could, then put lavender oil on it. I covered that with a scooby doo bandaid (this may have been the thing that helped lol) to absorb the blood. I kept it somewhat elevated, but I did finish making dinner.
After a couple hours, I changed the closure and the bandaid again, then put lavender on it again. (always after, or they wont stick!) The next day, I left it open, but with the closure on it. The third day I took the closure off completely. I still have a small scab hanging on there, but it should be completely healed within the next week.