In the spirit of TMTOWTDI, here's a version that only uses core Perl (ie. no Date::* modules):

#!/perl/bin/perl -w
use strict;
use Time::Local;

my $next_target_day    = 'Mon';
my $one_day            = 86400;

my @DOWname            = qw/ Sun Mon Tues Wed Thurs Fri Sat /;
my %DOWhash            = map { $DOWname[$_] => $_ } 0 .. $#DOWname;

my @T                  = localtime(time);

my $days_til_target    = $DOWhash{$next_target_day} - $T[6];
$days_til_target      += 7 if $days_til_target < 0;

my $beginning_of_today = timelocal(0,0,0,$T[3],$T[4],$T[5]);
print scalar localtime($beginning_of_today), "\n";
print scalar localtime($beginning_of_today + $days_til_target*$one_day
  + $one_day/2 ), "\n";
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