How do I write an Ansible handler with multiple tasks?

There is proper solution to this problem as of Ansible 2.2.

handlers can also “listen” to generic topics, and tasks can notify those topics as follows:

handlers:
    - name: restart memcached
      service: name=memcached state=restarted
      listen: "restart web services"
    - name: restart apache
      service: name=apache state=restarted
      listen: "restart web services"

tasks:
    - name: restart everything
      command: echo "this task will restart the web services"
      notify: "restart web services"

This use makes it much easier to trigger multiple handlers. It also decouples handlers from their names, making it easier to share handlers among playbooks and roles

Specifically to the question, this should work:

    - name: Check if restarted
      shell: check_is_started.sh
      register: result
      listen: Restart processes
    
    - name: Restart conditionally step 2
      service: name=service state=restarted
      when: result
      listen: Restart processes

and in the task, notify handlers via ‘Restart processes’

https://docs.ansible.com/ansible/latest/user_guide/playbooks_handlers.html#naming-handlers