As per ikegami's answer (Re: Types::DateTime, DateTimeUTC->plus_coercions( Format['ISO8601'] ), output format override question), you want to set a formatter on the DateTime object.

$cowtime->timestamp->set_formatter($formatter);
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But you probably want your CowTime class to be doing that automatically. You can do that with triggers:

use 5.12.0;
package CowTime {
    use Moo;
    use Types::DateTime -all;
    has timestamp =>
        is => "ro",
        isa => DateTimeUTC->plus_coercions( Format['ISO8601'] ),
        coerce => 1,
        trigger => sub {
          my ($self, $value) = @_;
          state $formatter = DateTime::Format::MyFormatter->new;
          $value->set_formatter($formatter);
        };
    1;
};
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The trigger is a coderef which gets called any time you set the value of an attribute. (Though it's not set if the attribute is set by a default/builder.)