片方 means one of a pair. But 片方の眉を吊り上げる、somehow sounds awkward to us, though there's no grammatical problem. We say simply 眉を吊り上げる when we are angry, or heard a shocking thing. This is possibly we Japanese don't have the idea of singular / plural forms of noun.
You cannot say 彼は左の手の片方の指を折れていた for "He broke one of the fingers on his left hand." You can say 彼は左の手の一本の指が折れていた, or 彼は左の手の一本の指を折った。But be aware that 一本の指が折れていた means status of his fingers, i.e. One of fingers of his left hand is broken , while 一本の指を折った refers to his action, i.e. He broke the bone of one of his fingers.
You cannot say '指を一本折れていた.' You should say '指が一本折れていた.'The function of noun, 指 in "指を" is an objective. 指 in "指が" is a subject.