I believe Koala meant discrete (instead of 'external'). that is, a PCIe add-in GPU (vs part of CPU).
Though with a really small PC, using Thunderbolt 3 or 4, and an external case holding a discrete GPU is an option. But realistically, at that point, from a size perspective, your probably better off getting a SFF or mini-tower instead
There are a number of Mini PCs that support a small form factor GPU. However, modern GPUs tend to be full height, so that limits how mini the PC can be and still support a discrete GPU. However they do exist, and I've seen numerous recent articles/blog/YT videos on such.
ex
https://www.servethehome.com/minisforum-ms-01-launched-10gbe-mini-pc-first-look/
note the PCIe slot is mechanically x16 but only x8 electrically. Then again, depends on what you are doing as that may not be an issue
Having researched this previously, I suspect your best path is to identify the GPU you want to use (from what I recall hearing, the cheapest AV1 encoder is one of the Intel ARC series). You'll want to research features that are important to you beyond AV1 encode offload from CPU (you probably one need 1 encoder... some higher end GPUs have dual encoders), then size and power requirements. Most Mini-PCs don't have a power supply not extra connection for a power hungry GPU. So power budget will be important, and then size/space constraints.
I suspect you'll want to come up with a list of low-power discrete GPUs, small in form factor, and then find a mini PC that will fit.
Servethehome.com has a whole 1L miniPC review going on. A number of their lower-end really small stuff won't apply to what you are looking for.
