You'll have to do two things.

(1) As pointed out by others, <SOMETHING> is ambiguous. It's either a filehandle read or a directory glob (to get the list of files that match a pattern:

$line = <$filehandle>; # simple scalar variable = read op
@files = <*.c>;        # anything else is a directory glob
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As the comments there say, the rule is: If it's a simple scalar variable, it's a filehandle. If it's a bare word it's a filehandle. Anything else and it's a directory glob.

So you'll need to use a temporary variable:

$tmp = $this->{THAT}->[0]->{THE_OTHER};
$line = <$tmp>;
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(2) You won't be able to successfully open more than one file (even if you hadn't hard-coded the filename :-). A common hack in Perl to get filehandles in arrays or hashes or subroutines is to use typeglobs, because Perl lets you use a scalar containing a typeglob anywhere you would normally use a filehandle. However, there's some cunning required when opening the file:

open FH, $file1 or die;    # open it, okay ...
push @filehandles, *FH;    # push the typeglob, fine ...
open FH, $file2 or die;    # reopening it, closing the previously open
+ed file!
push @filehandles, *FH;    # push it again
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This gives you two copies of the same glob in @filehandles. What you need to do is create a new glob each time. The best way to do this is with local():

sub open_file {
  my ($self, $filename) = @_;
  local *FH;

  open FH, $filename or die "opening $filename: $!";
  $self->{FH} = *FH;
}
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Good luck!

Nat