You probably want something like the following (not heavily tested):

use warnings;
use strict;

@ARGV == 2 or die;
my @files = @ARGV;

my $date_qr = qr{^(\d{4}/\d\d/\d\d \d\d:\d\d:\d\d\.\d{3}) };

my @dates;
for my $i (0,1) {
    open my $fh, '<', $files[$i]
        or die "Can't open $files[$i]: $!\n";
    while (<$fh>) {
        next if /Epoch/;
        chomp;
        m/$date_qr/ or die "$files[$i]:$.: invalid line: $_\n";
        $dates[$i] = $1;
        last;
    }
}

print "earliest=@dates\n";

my $i = 0;
if ($dates[0] ne $dates[1]) {
    my $earliest = $dates[0] lt $dates[1] ? 0 : 1;
    open my $fh, '<', $files[$earliest]
        or die "Can't open $files[$earliest]: $!\n";
    while (<$fh>) {
        next if /Epoch/;
        chomp;
        m/$date_qr/ or die "files[$earliest]:$.: invalid line: $_\n";
        last if $1 ge $dates[!$earliest];
        $i++;
    }
}

print "there are $i earlier date entries\n";
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Dave.