Shane,
mailgun.com is what I went with for staging's ESP. Requires a credit card to register but it's free up to 10,000 messages/month, only charged if you go over. You should make an account for production; it'll require some DNS record updates if you want better deliverability. (I set this up for aternum.dev, it takes about 4 or so hours for the DNS changes to take effect.)
When you submit a forgot password request, you'll now get an email with a link to reset it.