Handlebar mittens?

No experience, but was asking myself this very question the other day.

This US-made model is intended for drop bars or flat and gets good reviews around the Internet:

https://bikeiowa.myshopify.com/products/pogielites.

Hits the "gravel price" bracket too  

As far as I can see, drop-bar compatibility is achieved by installing them "backwards" (so the end normally closest to the stem on a flat bar bike is now down on the drops). Maybe this would be worth trying with some cheap ones sourced from large e-commerce company

Hand guards are a thing in the MTB world so it's only a matter of time before someone does a "gravel" version for drop bars and doubles the price

I use pogies right through the winter. An absolute godsend when having to use a smartphone regularly while out on the bike making deliveries. Unless it's extremely cold, I'll just use them without any additional glove underneath.

I know other people who can't get on with them, but I love 'em. Decent ones, aren't exactly cheap, but you do get what you pay for. It's Bar Mitts for me everytime.

I have often pondered whether it is possible to create a wind shield to hide fingers behind. Assuming you don't need your drops, a clamp on the lower section to a sturdy frame and then some plastic in front of the brake levers and above to the hoods might to the trick.

I ride with three pairs of gloves, merino liner, DHB thin glove with some sort of rubber backing then a winter glove, but biting winds still get through. Often I just want to hide my fingers somewhere, they'd be warm enough if sheltered.

Bar Mitts are great! They can be a bit too warm above freezing, but really come into their own below about 20 °F / -7 °C. I generally have a thin liner glove on under, which I often end off pulling off if my hands start to sweat.

No experience, but I did see someone on the south circular in London with some bar muffs fitted to a hybrid. They looked like std motorbike ones which have stiffeners for ease of entry and so the levers don't get applied at speed due to wind speed.

I never tried motorbike ones because my motorbike was kept outside. The idea of putting my hand in a cold, wet muff was just too much for me......