You can send transactional emails with the Laravel PHP framework using two methods: by setting up the Postmark adapter and sending email via SMTP. To set up the Postmark adapter, you need to create a Postmark developer account, sign up for a free trial, and then configure your Laravel project by installing the `coconutcraig/laravel-postmark` package, adding your Postmark token to the `config/services.php` file, and updating the `MAIL_MAILER` setting in the `.env` file. Alternatively, you can send email via SMTP using the Postmark SMTP tab, which requires logging into your Postmark dashboard, configuring the SMTP settings in the `config/mail.php` file, and setting up a sender signature. Both methods allow you to send transactional emails through your Postmark account.