Monitors are getting bigger, phones are getting smaller... how does a designer decide what screen(s) to optimize for?

I never start by thinking of screen size. Usually my initial PS files are in excess of 4000px by 4000px. The aim is to ensure the content is right, relevant and actually needed.

After that process, when looking to get it into a workable form I let the (potential) audience dictate. If the majority are on mobile, then start there and scale up. On desktop, scale down.

Unfortunately, with all the focus on mobile the sad fact is 27in & upwards have simply been neglected. Not a big enough audience.

Seems like this problem may end up becoming more pronounced. It feels like we're perpetually on the edge of the flat screen TV becoming another true internet device. Seems likely that will be an App-like device (a la iphone) as opposed to a browser-like device. But still it will challenge layouts even further.

I agree.

There also seems to be a great opportunity to really tackle multi-device interaction with media too. Which is lacking. For example, why can't we just use smartphones as a remote control to TV. Applying the strengths of each device together to bring in a whole new concept of entertainment and consumption.

Unfortunately most I have seen (that seem worthwhile) are conceptual. Here is an example of a multi-device concept – with TV. lab.syzygy.de

World of possibilities.

I also agree with all comments but will add, that taking a device-agnostic approach to any project is always the best way forward not thinking about the constraints of the technology (but be mindful of it) and design to target the audience with out diminishing the experience.

I find working from the most restrictive to the most indulgent and not to targeting any particular screen allows me flexibility, the experience is the same, but the more space the more indulgence (visually).