You heat the branding iron up to red hot and then sneak up on a cow and shove the branding iron on it's arse, then chase the cow down and cut the patch sized piece out with a box cutter and then run like hell.
All sorts of leather for patches can be used this way but just remember that some creatures (like ethnic people from Croydon) tend to get a bit uppity when stabbed in the arse with a red hot branding iron, and doing it to small things like rabbits and squirrels usually means their demise in a smoldering, screeching mess. And don't even TRY using octopus leather. They just explode.
Once you get really good at it you can up the game by making a leather jacket shaped cookie cutter and running after wildebeests at the local zoo after dark.
At least that's how we did it when I was a kid, only we used buffalo.