I solved this choppy audio issue with just one simple trick. (This might not work for everyone, but I solved it this way)

Short answer

If your hardware uses Bluetooth 4.0 instead of 4.2 or above, this solution is highly likely to work for you.

Open System Preferences > Sound > Input

Change sound input from Airpods to Internal Microphone

Long answer (+ Possible explanation)

(This is just my hypothesis based on my observation)

First of all, to clarify my symptoms, I am using a MacBook Pro 15 inch (Late 2013) which has Bluetooth 4.0 hardware. The choppy audio issue only happened in my MacBook Pro, and it did not happen in my other Apple devices such as my iPhone 7 Plus (which btw, has Bluetooth 4.2), nor my iPad Pro (also Bluetooth 4.2).

Based on these observations, my hypothesis is that Bluetooth 4.0 devices have too low bandwidth to handle both input/output (microphone & speaker) at a high quality. However, if I just change the audio input to the internal microphone, the Bluetooth connection only delivers sound output, so I think it can handle higher quality audio. On the other hand, if I use both microphone & speaker on a Bluetooth 4.0 device, I think the connection degrades and sacrifices audio output bandwidth in order to make room for delivering audio input.

Indeed, according to some sources, Bluetooth 4.2 is 250% faster and has 10x more capacity

Reference:

1. https://www.quora.com/What-are-the-main-differences-between-Bluetooth-4-0-4-1-and-4-2-in-the-Layers-Baseband-LMP-L2CAP-app-Layer
2. https://www.semiconductorstore.com/blog/2015/BLE-4-2-vs-BLE-4-1/1548/

After weeks of poking at this, I've managed to figure it out.

This actually had nothing to do with the AirPods or the external monitor, rather the problem was that Google Hangouts has its own settings to choose which devices are being used for input/output that overrides the system settings. So, while I was frantically trying to change which device was supplying in/out in the system preferences, Hangouts was happily sitting on the devices it had chosen, ignoring those changes.

Once I realized what was going on, changing the setting manually in Hangouts from my other wireless headphones to the AirPods proves effective every time.