Could there have been two "Big Bangs"?

I believe you are referring to the phenomenon known as 'inflation'. It refers to the extremely rapid expansion of the universe shortly after the Big Bang. According to wikipedia (linked above) it lasted from $10^{−36}$ seconds after the Big Bang to sometime between $10^{−33}$ and $10^{−32}$ seconds. Amongst other things, it helps explain why the cosmic microwave background looks as it is observed.

While I can't seem to find a reference for it, I seem to remember the "double expansion" was a way to explain the paradox that the background radiation was too uniform for stars and clusters to form within the time frame allowed between the start of the expansion and now.

If I remember correctly the theory was the initial cosmic background radiation spread out, and a second expansion started the unevenness that allowed clusters and stars to form.

Big-Bang proponents theorize that small clumps of gas in their primordial fireball grew into stars, galaxies, and clusters of galaxies. The clumps should have left evidence of themselves as hot and cold spots in the cosmic microwave background. That is, as we point our detectors in different directions, we should see slight differences in the temperature of the microwaves from point to point in the sky.

With better measurements we now know that the cosmic background radiation is uneven which conforms better to our expectation of the time it took stars and clusters to evolve. So there's no longer a need for a more complicated theory.

However the jury is still out on the distribution of matter vs. radiation. Several decades ago two Soviet astrophysicists, R. A. Sunyaev and Ya. B. Zel'dovich, pointed out that as the background radiation passes through large clouds of intergalactic gas, some of the radiation would collide with electrons in the gas, scattering it out of our line of sight and giving it a different wavelength ("Compton scattering"). The resulting change of intensity in the background radiation reaching us would be interpreted a change in the radiation's temperature.

Likewise we are still in the process of trying to measure the CBM and if gravity waves are present from those early days. If this next generation Planck Telescope shows that there is no obvious distortions caused by gravity waves, it could rule out the Big Bang plus inflation theory -an add-on theory that explains the phenomenal sudden expansion of space from a tiny point.