The Mooseish way to get the dynamic class name is $obj->meta->name:

use 5.010;

package Parent;
use Moose;
sub who_am_I { shift->meta->name };

package Child;
use Moose;
extends 'Parent';

package main;
say Child->new->who_am_I;
__END__
Child
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$obj->meta returns the metaclass instance for the object's class, and ->name returns the name (surprise, surprise).

(excursion: In Perl 6 it works basically the same, except that there is a shortcut for calling methods on the metaclass: say 1.^name says Int).