No, BASH syntax doesn't allow any kind of nesting. You can do so using external utilities like cut:
cut
date | cut -c 1-10
Wed Jun 13
To replace date:
date
$ dateWed Jun 13 14:57:38 EEST 2018
you can use printf:
printf
$ printf "%(%a %b %d%n)T"Wed Jun 13
man strftime for more format modifiers.
man strftime
If you want the output Wed Jun 13 for today's date (which this is), then
$ date +'%a %b %e'Wed Jun 13
See the manual for date and/or for strftime on your system.
strftime