Are there any programmes to identify modular "NeuroBricks"? In synthetic biology, an organization called the BioBricks Foundation tries to identify modular biological components that are amenable to engineering design, and publishes them in the Registry of Standard Biological Parts. Does an analogous movement/organization/research group exist in the neuroscience community working on finding general "NeuroBricks"? These could come in the form of either individual neurons with well-defined abstracted functions or neural networks that do predictable things (i.e. adders, logic gates, amplifiers, etc.)

I ask because I know that the field of neuroengineering is a thing, but I have no idea how possible it is to do direct engineering of neural networks or neurons (it seems like the current consensus is somewhere in the range of really really hard to impossible).
Assuming that by "neurobricks" you imply the ability to design systems using modular neuronal components, then I think the Neurological Engineering Framework (NEF) and Nengo, the software used to leverage the NEF, is what you're looking for.

Essentially, groups of neurons fire together as a way to encode information from various sources. So you can get simple things like an integrator or something that keeps track of where to move a joystick (like in the research by Georgopolous), but that can scale up (because it's just a group of neurons) all the way to a complex model of the brain like Spaun.

In terms of engineering applications, there's a lot of things that these neurons are particularly good at, such as robotic arms, as shown in Spaun.