There are a bunch of different types of gpu bracket. Some of them come with GPUs, some of them don’t. Some are metal, some are plastic some are even rods somewhat resembling chemistry glassware stands that magnetically attach to the inside bottom of the case. At question here may be the term “gpu bracket” which most commonly refers to braces that keep heavy GPUs from sagging and putting undue pressure on the motherboard pcie connector.
what you might be talking about is a pcie slot attachment slot. If this is the case (as a 1060 mini is unlikely to need a gpu bracket as I understand the term) it likely doesn’t matter a ton which way it is done. Most commonly one would attach the case screw first then the gpu screw(s there should be two on a double slot card) but if getting the thing to fit is an issue attaching the back plate to the gpu first then attaching the whole thing to the case is probably fine. Not all cases have the sort of pice slot system you have pictured. It’s more common in some smaller cases but even then is not ubiquitous.