How to build a high sensitivity EEG headset for continual monitoring?
There are a few commercial EEG makers that do allow monitoring of raw EEG data. For instance, I know the Muse headband and OpenBCI allow you to do this with little trouble. These both support sampling rates up to 250 Hz. (The OpenBCI in principle could have a higher sampling rate, but i don't know anyone who has done this.)

The Emotiv EPOC also has an option to get raw EEG data. You have to pay more for the research version of the SDK to get the data officially. However, someone has reverse engineered their protocol, so you can access the raw data that way. This has a sampling rate of up to 128 Hz.

That said, if you still want to build your own EEG, OpenBCI has documented all of their plans, so a good start would be to replicate their product. 

There is also the older, but still useful, OpenEEG project, which also has open plans for an EEG device.