#perl
use strict;
use warnings;

sub escape {
    my( $str ) = @_;
    $str =~ s!<!&lt;!g;
    $str
}

while( <DATA>) {
    s!(?:^|\G)([^`\\]* # plain string without escape or backtick
               |\\.    # anything, escaped
               )*
               (`      # backtick
                 (?:[^\\`]*|\\.) # only escaped backticks
                `)     # closing backtick
     !$1.escape($2)!sgex;
    print
};

__DATA__
This is `my <string>` that I want to modify because the angle bracket is between backticks
This is `my &lt;string>` that I want to modify because the angle bracket is between backticks
This is `a <string> I don't want to modify because it's not between backticks
This is \`another <left_angle_bracket>`I don't want to modify because the first backtick is escaped but I do want to modify <the_last_left_angle_bracket>` between the backticks