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"something resembling the handle of a pan," 1851, from pan (n.) + handle (n.). Especially in reference to geography, originally American English, from 1856, in reference to Virginia (now West Virginia; Florida, Texas, Idaho, Oklahoma, and Alaska also have them). Meaning "an act of begging" is attested from 1849, perhaps from notion of arm stuck out like a panhandle, or of one who handles a (beggar's) pan.
"to beg," 1888, from panhandle (n.) in the begging sense. Related: Panhandled; panhandling.
To beg, esp by accosting people on the street: The boys deal drugs or panhandle, even become male prostitutes
[1903+; fr panhandler]