Can I power a Raspberry Pi 3 + Accessories off of a Laptop Battery Cell?

I am a bit of a beginner when it comes to electrical engineering, I know the basics but not all that much more than that. I am working on a portable Raspberry Pi project and I'm attempting to figure out a safe, small and relatively cheap portable power option.

I've got one of these battery packs lying around from a laptop that doesn't work anymore. Could I pop it open and attach one or two of the cells to one of these types of "Power Bank Charging Boards" and just use output to power the components and simply charge it via usb? Will I be able to drain the cells and charge them at the same time?