First, keep in mind that since this is a hash, order is arbtrary so there is no "first" file name. If order matters, you should be using an array of arrays rather than a hash of hashes.

Since you are dealing with an object of unknown depth, a recursive parser is likely easiest. Maybe something like

#!/usr/bin/perl
use strict;
use warnings;

my $VAR1 = {
          'File_Handler' => {
                                 'Compile_File' => undef,
                                 'Log_Details' => undef,
                                 'Complete_Logging' => undef,
                                 'Get_Config' => undef,
                                 'Get_Data' => undef,
                                 'Get_Abc_Data' => undef,
                                 'Internal_Operations' => {
                                                     'Get_Abc_Data' =>
+ undef,
                                                     'Start_Operations
+' => undef,
                                                     'Get_Start_Up_Dat
+a' => undef,
                                                     'Construct_Failur
+e_Ident' => undef,
                                                     'Construct_Additi
+onal_Words' => undef,
                                                   },
                                 'Start_File_Handler' => undef
                               }
        };

sub output_files {
    my $path = shift;
    my %hash = @_;
    for my $key (keys %hash) {
        if (ref $hash{$key}) {
            output_files("$path.$key",%{$hash{$key}});
        } else {
            print "$path.$key.2.ada\n"
        }
    }
}

while (my($key, $value) = each %$VAR1) {
    output_files($key, %$value);
}
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