For sure, the Dollmaker understood that, like the Mona Lisa,

 it was the SMILE that bespoke the Icon...

hmmm... but she still wondered about the back-story...

A Stepped-up Investigation of  

Image
 

What inspired Lewis Carroll?

Did he Know any Smiling Cats?

For lovers of language, the imagination, and feline mischief

How deeply did Carroll intend its meaning to run?

And!

If they DO smile... WHY?

Hard questions...

Yet, hotly debated anyway,

 like so many Other Questions {not only for deceased authors} which shall never be answered...

 but the Dollmaker digresses...

Curious Minds are driven to ask:

What is really going on with this mysterious Smiling Cat?

Here are the Clues...

Possibility #1

Regarding the Issue of the Inspiration for the Cheshire Cat being a Real Cat:

The British Shorthair is the most likely suspect

the habitat of this breed being: Cheshire County, England!

Since Lewis Carroll grew up in Cheshire County,

he might have even had one of these cats as a pet!

And Some DO Say: This breed of Cat has a natural Smile.

The British Shorthair is a well-balanced, medium to large, sturdy cat

with a powerful body, a broad chest, short, strong, substantial legs and a thick tail.

 Round is the key word for the British Shorthair, accurately describing the head, cheeks, eyes and paws.

 One of the most appealing features of the British Shorthair is their built-in smile caused by their round whisker pads.

 The smile is like Lewis Carroll’s Cheshire cat; originally a tabby British Shorthair.

-wikipedia

Cheshire, England

(could pass for Diagon Alley in the Harry Potter series..)

• Cheshire Cheese shaped like a Cat:
•  In England, the County of Cheshire
• is known for its dairy production,
•  particularly its cheese.
• Carroll grew up in the Cheshire village of Daresbury,
• where he no doubt would have been familiar
• with the local practice of producing cheeses
• in the shape of a grinning cat.
• The cheese was eaten from the tail to the head,
• leaving the smile for last. (aha!)
• 
• Possibility #3

• These Cats are just plain Happy: 
• In the Port Town of Chester, (in Cheshire County)
• there’s a local legend set around the Docks.
• It’s said that Cheshire County Cats SMILE because
• they are the happiest Cats in all of England!
• They are not only present for all that dairy production...
• but they also get to feast upon wayfaring rodentia
• lured into the town from docked boats
•  by the scent of milk and cheese.

• The Cat Carving at St. Wilfrid's Church:
•  
• In the Cheshire village of Grappenhall, there’s an old Church
•  which has been there since the 12th century.
•  
•  Young Lewis Carroll's father, a vicar, used to preach there (!)
• and
•  
• A carving of a grinning cat
• peeks out from above the church’s main entrance.
•  

• So, It would seem as though this was not really an uncommon image
•  for Lewis Carroll as a boy.

Therefore... the safest and probably most accurate Speculation would seem to be:

"All of the above."

and that sounds about right to the Dollmaker too....