Place eggs in a pot, cover with water and bring to a full boil. Turn off heat, cover pot and let stand 10 minutes. Drain water and crack eggs by shaking pot. Run eggs under cold water until cool enough to handle.
Peel eggs and trim tops. Remove yolks to present eggs upright. You should have 12 upright eggs.
To the hard-boiled egg yolks, add grated onion, grating directly over bowl to catch the onion juice, garlic, relish, Worcestershire, mustard, hot sauce, salt, pepper, paprika, mayo – a few tablespoons to start – mash to very smooth, adding a bit more mayo if too dry.