... I didn't have a hope of figuring out the RE on my own...

This problem is actually used as an an example of the use of (?R) and friends in perlre:

Win8 Strawberry 5.30.3.1 (64) Thu 10/01/2020 19:06:47
C:\@Work\Perl\monks >perl -Mstrict -Mwarnings
use 5.010; # need 5.10+ extended patterns (?R)

use Data::Dump qw(dd);

my $str = '(a a (a a)(a a))(b b(b)b b) xxx (c (c (c c)))';

my $rx_bal_parens = qr{
    \(
      (                # Capture group 1 (contents of parens)
        (?:
          (?> [^()]+ ) # Non-parens without backtracking
        |
          (?R)         # Recurse to start of regex
        )*
      )
    \)
    }xms;

my @bal_parens = $str =~ m{ $rx_bal_parens }xmsg;

dd \@bal_parens;
^Z
["a a (a a)(a a)", "b b(b)b b", "c (c (c c))"]
[download]

(Of course, if you use something like this, do thorough testing to be sure it really does what you want. :)

Update: Fixed first line of code (the use 5.010; statement). It somehow got stuck onto the end of the command line during cut-pasting.