When I need to empty my inventory of junk, I use Control Q to remove stacks at a time but when I use Control Q on an empty slot by accident, it drops the mining gadget if it is in my hand. It won't drop if the gadget is in my hotbar nor in my inventory.
TLDR:
If Mining gadget is in hand AND I hit Control Q on an empty slot in my inventory Then drop the Gadget.
Thank you for your great work!

This is actually a forge bug not ours. I talked to the forge guys regarding this issue and apparently it can be fixed but just hasn't been fixed yet. I've lost my gadget on ForgeCraft once or twice it's pretty annoying but if you know about it you can avoid doing it. If you want to open an issue on forge / check on the forge issue list for the bug and reference the issue here I can let you know if it get resolved or not / possibly fix it myself maybe... :P