"to slice or cut (a finger)" in Traditional Chinese / Spoken Cantonese?
I would say that "開面" as "to slice open" in cantodict is wrong. Have a look at 國語辭典, these two terms are listed:

http://dict.revised.moe.edu.tw/cgi-bin/cbdic/gsweb.cgi?o=dcbdic&amp;searchid=Z00000076573

http://dict.revised.moe.edu.tw/cgi-bin/cbdic/gsweb.cgi?o=dcbdic&amp;searchid=Z00000099591

About "I sliced my finger", you may consider:  (u+20779). I guess that your computer / tablet needs to have a suitable font in order to display it.



Usage:
Your colleague asked "What happened to your finger?"

Answer: "比張紙親" (Click each character for its sound file :)

Roughly, "比張紙" is "by a piece of paper"; "親" is "sliced".

Be careful :)

Edited.

Of course you can :)

"比" (u+6bd4), or "俾" (u+4ffe) are equal in this context, you can use either one, I quoted "比" just because it's simpler, has fewer strokes.

Now, here's another one: "畀" (u+7540), which is the ancestor of the above two. It existed since oracle bone script, used commonly in literatures of yore. Unfortunately, nowadays this original character is not well recognised; that's why people are using "比" or "俾" most of the time.