The Brenizer Method is simply taking a grid of photos of a reasonably close portrait subject with a long or longish focal length and wide open (or close to it), and then stitching them together to make a single final portrait image.
What it gives you is an image that looks wide or wide-ish and still has a shallow depth of field. Usually it has a depth of field more shallow than you would expect for that lens and format. It is a great way to mimic medium or large format results.
It was popularised in recent times by the work of Ryan Brenizer, hence the name. It is also known as panoramic stitching, bokeh panorama and bokehrama. The actual technique or idea has been around for ages – since 1843 according to the Wikipedia entry.
Anyway, I’ve been playing with it for a while now. Here’s two recent-ish shots. With both more and less attention to fixing up the background oddness that can result. In fact, I often like the surreal weirdness that can occur after the stitching.