How do I wash farm-fresh eggs?

Do nothing.

In Europe (Germany at least), eggs aren't washed at all.

Store them in a dry place, no fridge required.

My personal suggestion would be to immediately before the egg is used

1. Using a clean dry brush try to remove any obvious dirt in a ring around the egg where you are trying to break it open.
2. Next clean that area with a dilute bleach or milton solution.
3. In using the egg, you may then wish to comply with FDA safe handling instructions and cook the eggs until the yolk is firm.

• Safer still: politely accept the eggs, bin them and substitute for retail eggs.

My Reasoning

Regulations and procedures may vary between countries, but the science and potential risks depend on the procedures used.

It seems to me your idea of trying to remove and sanitise the shell before cracking it open must be better than cracking an obviously contaminated shell as it is.

As far as I know in my country {England & no longer part of the EU} it is not permitted to wash eggs that are to be made available to the public in their shells, but I can not seem to find evidence to support that conjecture other than this

National FSA advice ... Temperature changes can lead to condensation on the surface, which causes increased penetration of Salmonella from the outside of the shell into the egg. ....

That serves as a reminder that even pristine looking retail eggs are potentially dangerous, and risk contaminating food & if washed; or placed in a fridge; the egg within the shell.

And Ireland's regulations {A European country within the EU} say it is illegal to wash Grade A hens eggs.

https://www.fsai.ie/faq/egg_washing.html ... The egg marketing rules stipulate that grade A hen eggs may not be washed. Only grade B hen eggs may be washed and these cannot be sold at retail, ... Is washing of eggs recommended?

No, because washing may aid the transfer of harmful bacteria like Salmonella from the outside to the inside of the egg.

The priority in egg production is to produce clean eggs at the point of collection, rather than trying to clean them afterwards. The cleanliness of the egg should be assured by good management and hygiene of the poultry house, and by minimising the delay between egg laying and egg collection.

In Shell Pasturisation

Not common in my country. But it appears in the USA some of your retail eggs may be carefully washed, pasteurised in shell { See https://en.wikipedia.org/wiki/Pasteurized_eggs#Pasteurized_shell_eggs } Then after pasteurization, the eggs are coated with food-grade wax to maintain freshness and prevent environmental contamination and stamped with a blue or red "P" in a circle to distinguish them from unpasteurized eggs.

Wax coated eggs would presumably be fine rinsed under water, or stored in a fridge.