That ain't what Text::Balanced does.

Text::Balanced::extract_bracketed( $str, '[]' );
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... will extract the first (and only the first) [...] delimited string, allowing balanced pairs of square brackets within it. It doesn't attempt to then do any parsing within that string. It will also return the remainder of the input string, so you can parse that remainder to, for example, find the next [...] delimited string.

If you need deep parsing, then you need to write a recursive function. Something like this:

use strict;
use warnings;
use Text::Balanced qw(extract_quotelike);
use Data::Dumper;

sub parse_str
{
    my $str = shift;
    $str =~ s/\A\s*//;
    
    # list-like
    if ($str =~ /\A\s*\[/)
    {
        # eat initial [
        $str =~ s/\A\s*\[\s*//;
        
        my @arr;
        while ($str !~ /\A\s*\]/)
        {
            my ($term, $remainder) = parse_str($str);
            push @arr, $term;
            if ($remainder =~ /\A\s*[,\]]/)
            {
                $str = $remainder;
                $str =~ s/\A\s*,\s*//;
            }
            else
            {
                die "Expected comma; got <<<$remainder>>>";
            }
        }
        
        # eat trailing ]
        $str =~ s/\A\s*\]\s*//;
        
        return \@arr, $str;
    }
    
    # quote-like
    elsif ($str =~ /\A['"]/)
    {
        my ($term, $remainder) = extract_quotelike($str);
        $term =~ s/\A['"]//;
        $term =~ s/['"]\z//;
        return ($term, $remainder);
    }
    
    # number-like or bareword-like
    elsif ($str =~ /\A([\w\.]+)/)
    {
        my $term = $1;
        return ($term, substr($str, length($term)));
    }
    
    die "Expected something else; got <<<$str>>>";
}

print Dumper parse_str <<DATA;
[
    'listliteral',
    [
        'ListLiteral',
        [
            '../classSPL_1_1Operator_1_1Instance_1_1ExpressionTree_1_1
+ListLiteral.html',
            1,
            'SPL::Operator::Instance::ExpressionTree'
        ]
    ]
]
DATA
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