When your new blog posts are published from your ButterCMS account, you can let people in your mailing lists know about it by using Mailchimp.
All you need to do is to set up a webhook (https://buttercms.com/docs/api-client/other#OverviewWebhooks) which will notify your system of a new blog post.
Your system would then ping the Mailchimp API to
1) Draft the new email campaign by pulling blog post content from Butter and Writing it to Mailchimp
2) Send out a new email campaign.