small workaround:
when you save the project file in the first place, don't give it a filetype.

ie instead of saving myproject.sublime-project,
save your project file as myproject.

Sublime will still recognise it as a project file.

Not desireably though because ST won't be linked to those .sublime-project files then because they obviously don't have that extension in the first place.

Where can I find - is there some API for switching projects dialog? Perhaps
I can write some small plugin...

Didn't find such API here: http://www.sublimetext.com/docs/3/api_reference.html

What do you want to do with the API? You can only open the dialog by calling the corresponding command.

Modify the way that dialog returns and display data.

Not possible.

1. Save the project file somewhere without a filetype e.g. ../Sublime/Project Files/kis
2. In Sublime Text, goto menu: Project - Open Project...
3. Navigate to ../Sublime/Project Files/ and double-click on kis
4. Anytime you press Alt+Ctrl+P, the project will be listed

Closing as a duplicate of the more precise issue #2353.