How do they produce an impedance/phase plot for a speaker with a chirp that lasts 1 sec?
Quoting Wikipedia article on Electrical impedance:


  Impedance is defined as the frequency domain ratio of the voltage to the current.


You can measure instantaneous current indirectly by connecting a small resistor (say 0.1 ohm) in series with the speaker and by measuring the instantaneous voltage across it, as function of time. Because the resistor is purely resistive, by Ohm's law $I = V/R$. To know the instantaneous voltage seen by the speaker, subtract the instantaneous voltage drop across the resistor from the instantaneous amplifier output voltage, which you can also measure as function of time. 


Figure 1. Impedance curve test circuit. Voltage is to be measured at the test points TP2 and TP3 referenced to TP1.

In order to do the frequency domain division, play back the test signal, measure both instantaneous voltage and instantaneous current across the speaker as function of time, digitize those two signals, do a discrete Fourier transform on both, and complex divide bin by bin in the frequency domain. As long as there are no bins that are (too close to) zero-valued, the result should be meaningful. A short chirp, even a click, would work as long as it contains all frequencies of interest. The discrete Fourier transform should be long enough to fit the portion of the current signal where "things are happening". That can be significantly longer than the chirp, if it is short.

This kind of testing relies on the assumption that the system is linear. Some causes of nonlinearities (mainly at the speaker if the amplifier has good specs) can be avoided by limiting the peak amplitude of the test signal. A chirp is better test signal than an impulse, because it carries more power for the same peak amplitude, improving the signal-to-noise ratio.