If you have a model that says the brain is a singular entity there will always be problems how to phrase meta-activity such as self-awareness and consciousness. If the brain is singular, what does the observing? Does the observing part belong to the brain or not? If the brain is observing itself, which part does the observing and which is the observed? Can we change the roles, and make the observee the observer? How does "the brain" decide what "the brain" is thinking about?
The problem with this model is that it's regarding the brain as a whole as if it's a single processing entity, since that's how we experience it, but physically that's far from the truth: the brain is a highly segmented parallel processing machine and (for as far as I know) there is no clear control hierarchy within the brain to point at a single location to say "this is where [thought/sensation/self] originates".
So my theory is that consciousness is the emergent property of many separate specialized brain areas responding to and interacting with each other: either directly (neuronal links), via the body (flexing a muscle tightens the skin, creating a sensory feedback loop), and through the world (turning your head changes what your eyes perceive).
So speaking in terms of functionality, there is no "brain" as a single entity. The brain is like an ant colony, using hormones and electrical impulses to keep its disparate parts acting in unity. Under this model, consciousness can be defined as the push and pull between these parts, and self-awareness as the consensus model under which the separate parts reached some equilibrium. You won't have problems defining which parts do the observing and which parts are observed: what we "experience" as our mind is the tug-of-war between the processing areas, not the processor activities themselves. And similarly, what we experience as consciousness is the ability of our "mind" to drown out that cacophony and purposefully cede control ("focus") to different brain areas at different times.
Of course, there's no evidence that any of this is true, and there's still plenty of questions about the mechanism through which this would happen, but for me personally this model has allowed me to move on from the vicious cycle of having to first define "brain" in order to define "thought" in order to define "awareness" in order to define "mind" in order to define "self" in order to define "brain".