Docker CMD exec-form for multiple command execution
The short answer is, you cannot chain together commands in the exec form.
&& is a function of the shell, which is used to chain commands together. In fact, when you use this syntax in a Dockerfile, you are actually leveraging the shell functionality.
If you want to have multiple commands with the exec form, then you have do use the exec form to invoke the shell as follows...
CMD ["sh","-c","mkdir -p ~/my/new/directory/ && cd ~/my/new/directory && touch new.file"]