I'd use the following, but Twig's limited XPath support cannot handle it:

use strict;
use warnings;

use XML::Twig qw( );

my $twig = XML::Twig->new(
   twig_roots  => {
      'Application[Properties/Timeout]' => sub {
         my ($t, $elt) = @_;
         print("$elt->{att}{Name}\n");
         $t->purge;
      }
   },
);
$twig->parsefile('a.xml');
[download]

This works:

use strict;
use warnings;

use XML::Twig qw( );

my $twig = XML::Twig->new(
   twig_handlers => {
      'Application/Properties/Timeout' => sub {
         my ($t, $elt) = @_;
         print($elt->parent->parent->att('Name'), "\n");
      },
   },
);
$twig->parsefile('a.xml');
[download]