Lacking a hand or a foot, a cook could not climb ropes, but he could stir the pot.
Notice the word lacking. The verb form has been changed from a past tense to a participle. You have to be very careful with this construction. My favorite troublesome sentence was one my high school English teacher used to use to warn us about dangling participles. (The sentence you have is OK).
Running up a tree, I saw a squirrel.
Who is running up the tree? If it really was you, then the sentence is correct. If you mean the squirrel, which is much more likely, then the participle is completely wrong. I saw a squirrel running up a tree. Much better.