Depends how you define "single exposure". If we're talking about a single lens projecting light onto a single sensor this is a physical impossibility. With a DSLR or a (regular) smartphone, you always end up with multiple pictures being stitched together.
But there are cameras out there specifically for 360° pictures that come with multiple lenses and multiple sensors, so the moment you take a picture you actually take multiple pictures that are stitched together automatically. The Samsung Galaxy Gear 360 is an example for this kind of thing that can be combined with VR-Gear which - in my eyes - is a sure way of getting sea sickness ;-)
That particular picture might be done by setting the timer of the camera and throwing it up. There are cameras that are intended to be thrown into the air in order to do 360°-panos, e.g. this one.