Does the time difference between Beijing and Chicago change during the year?

Yes. Because of daylight saving time the gap moves between +13h and +14h.

When is the best time to call Chicago from Beijing?

There is no overlap between standard working days, so one side has to call early or late.

Beijing is 13h ahead of Chicago. This gap is not fixed: daylight saving time moves it between +13h and +14h during the year.

Beijing

China

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Chicago

United States

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A 9-to-5 working day in Beijing does not overlap one in Chicago at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Beijing · China	Asia/Shanghai	UTC+8
Chicago · United States	America/Chicago	UTC−5

What is the time difference between Beijing and Chicago?: Beijing is 13h ahead of Chicago. This gap is not fixed: daylight saving time moves it between +13h and +14h during the year.
Does the time difference between Beijing and Chicago change during the year?: Yes. Because of daylight saving time the gap moves between +13h and +14h.
When is the best time to call Chicago from Beijing?: There is no overlap between standard working days, so one side has to call early or late.