It's mostly about lighting from an angle well off the camera lens' optical axis combined with a very wide aperture to get narrow depth of field.
Here the key light is fairly hard from above and wide to camera right, with much weaker and softer fill almost directly to the left of the subject.
In the case of this portrait, the effect is even more noticeable because the focus is somewhere between the tip of the nose and the lips, rather than the eyelids that are further away from the camera. There also seems to be a hint of tilt involved, as the field of focus seems to be slight further back on the right side and slightly further forward on the left.
Based on the date of the portrait as well as the appearance of this and other works by Will Burgdorf, it was probably taken with a large format camera. Most LF cameras at that time were what we now call "view" cameras with a bellows between the front lens board and the rear film holder. This allowed for independent control of tilt and shift movements of the lens. It may be that the slight tilt seen in this self portrait is intentional, but probably more likely that the camera was just not carefully adjusted to insure the lens board was exactly parallel to the film holder. Or it could have been that the film itself was not laying properly flat against the film holder.