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:
About "I sliced my finger", you may consider: 𠝹 (u+20779). I guess that your computer / tablet needs to have a suitable font in order to display it.
Your colleague asked "What happened to your finger?"
Answer: "比張紙𠝹親" (Click each character for its sound file :)
Roughly, "比張紙" is "by a piece of paper"; "𠝹親" is "sliced".
Be careful :)
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.