"." is string concatenation. Not what you want.

But this is Perl. If you really, really want it, it can happen!

use 5.010;
use strict;
use warnings;

my $x = {foo => "abc"};

{
    package ChocolateSauce;
    use overload '.' => \&whipped_cream;

    sub whipped_cream {
        my ($adam, $eve) = @_[$_[2] ? (1, 0) : (0, 1)];
        $$adam{$_} = $$eve{$_} for keys %$eve;
        $adam;
    }
}

bless $x => 'ChocolateSauce';
$x .= {bar => 'def', baz => 'ghi'};

while (my ($k, $v) = each %$x) {
    say "$k => $v";
}
__END__
bar => def
baz => ghi
foo => abc
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