You need to set up your replacement string correctly. Once that's done it runs fine with the /ee modifier:

use strict;
use warnings;

use Test::More tests => 1;                      

my $new     = 'blah';
my $string  = 'asdfjkjl';
my $find    = '(asd)';
my $replace = qq#\${1} . '$new'#;

my $result = replace ($string, $find, $replace);

is $result, 'asdblahfjkjl';

sub replace {
    my $string  = shift;
    my $left    = shift;
    my $replace = shift;
    $string =~ s/$left/$replace/ee;
    return $string;
}
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