A thin leg resembling a broom handle, or historically, a person with thin spindly legs.
From broom + shank (Old English 'sceanca', the leg or shaft of something). Shank often refers to the leg, making this a compound describing leg appearance.
This is a delightfully obscure insult—calling someone 'broomshank' meant they had skinny legs like a broom handle, which shows how old English used everyday objects to describe people's physical characteristics.
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