🚀 go-pugleaf

Well, assuming that you mean the restaurant/hamburger company, it 
started in business a few years before there were computers or 
digitized fonts, in 1921, when the origianl logo was designed. So it 
certainly didn't use "a font" to do its signage, but was probably 
created manually.

Generically, it's a very plain Blackletter font, with very few 
embellishments.

Looking at images of the White Castle logo and advertisements, it 
appears in a number of slightly different typestyles, with varying 
amounts of embellishments (additional lines, diamonds, and ornamental 
additions).  To see the examples, Google/Images for:
"White Castle" Logo

I couldn't find a single blackletter font that comes close to all of 
the letters, but there might be one out there. Try googling for
Blackletter fonts to see if there's one that's close enough for 
whatever it is you want to use it for.

  - Herb