What you have there is a foil deboss. I work in litho print and have done a little research into foil blocking to see if it is possible to produce short runs or samples for clients. I have discussed the topic with professional foil blockers but have not found a cheap way to produce the effect you are looking for.
The professional method involves creating a metal plate that is heated on a special press. When the hot plate is applied to the substrate it debosses and foils as part of the same process. As you might imagine, creating this custom plate is expensive. There are other methods of foiling involving UV setting adhesives and foil film but these will not get you the deboss… and they are impractical on short runs as again, the machinery used is incredibly expensive and the cost is in the initial set up.
There are a number of methods of producing an emboss or deboss at home. You can laminate cut card layers then place a material over this, then use an embossing tool to work the detail into it, but from there you’re going to have to manually apply the gold, either with glue and foil or paint of some kind… which is time consuming and hard to make look professional.
Having spoken to people on this forum I have discovered it is possible to fuse gold leaf onto laser printer toner with judicious use of an iron, but of course you need to set up your artwork carefully so the areas you are foiling are relatively isolated. In theory you can then print over/around this with an inkjet printer… but this doesn’t help you with your current project.
There are two more things I can throw into the mix for you. One is a type of gold paint used by guilders which is effectively gold in solution. It might be easier to manipulate than sheets of leaf but maintains that bright shine. Perhaps if you can find a good method of masking the surface this might be effective. Perhaps screen printing can help you achieve this?
The other is gold embossing powder. This adheres to a mildly sticky glue and can be melted into place with the hot air rising fro a toaster. It’s popular with crafty types… but the interesting thing is that the glue can be applied with a rubber stamp, the powder sprinkled over the substrate, the excess gently blown away, and then heated to set the design. It’s possible to produce some quite fine work using this method, and it is easily repeatable allowing the production of short runs. You just need to create an appropriate rubber stamp.
This probably hasn't been much help but good luck with your project, and let us know what you discover in the process.