In the blog post by Gleb Bahmutov, the process of conducting full-cycle testing of HTML emails using SendGrid and Ethereal accounts is detailed. The testing involves sending emails through the SendGrid SMTP server and retrieving them from temporary Ethereal email accounts to ensure proper formatting and functionality. This process is integrated into a Next.js web application that requires users to register by confirming a code sent via email. The testing uses Cypress Test Runner to automate the end-to-end process, verifying that emails are correctly sent and received, and ensuring that the confirmation code is accurate and functional. To enhance the test's reliability, a retry mechanism is implemented to handle potential delays in email delivery, using the cypress-recurse plugin to repeatedly check for the email's arrival without relying on hardcoded wait times. This approach increases the robustness and efficiency of the testing process, allowing for a more seamless and dependable confirmation email flow.