Okay!
Create a New Effect Project — generic HD.
Turn off the Group containing the Effect Source.
Share/export > Save Current Frame
In the export dialog, select Settings then Export: PNG Image and Color Channels: Color & Alpha.
Name it thumbnail.png.
Turn the Group containing the Effect Source back on.
Go into the Finder and find your export — drag it back into the project and drop it on the Group. It should end up inside the group directly over the Effect Source.
Select thumbnail.png in the Layers list and select the Image tab and for Type: select Drop Zone.
Publish: Pan, Scale, Fill Opaque and Fill Color. [The drop zone is automatically published.]
Save the project as "Project Thumbnail". Provide a category that will make it easy to find.
In Final Cut: add an Adjustment layer at the middle of your project and apply Project Thumbnail effect to the Adjustment Layer. Pick a Frame from your Storyline (there's about a 90% chance the dropzone will be a still image.) You can use a selection from a clip in the Event Browser, but it will "play" if it is a video clip — you might need to reposition the adjustment layer slightly.
Click in the Event Browser — not on the current project — wait a few seconds until the thumbnail updates.
Applying the effect to an adjustment layer makes it "portable". If your project changes length, just reposition it at the new center. You can type "V" to toggle its visiblity if you need to export your project.
Tip: Select the Adjustment layer and right click on its title > select Rename clip and call it project thumbnail — it will make it easier to find with the Timeline index Search.
You can use the Pan and Scale feature to enlarge a section for clarification or you can Scale smaller, use the Fill Opaque and Fill Color "feature" to create a color outline for the thumbnail.
HTH — I may get back later with the WHY I chose this method.