Dolt's SQL server runs on a fork of Vitess. Vitess is a MySQL database sharding solution that was created to scale YouTube:
Vitess is a database clustering system for horizontal scaling of MySQL through generalized sharding. Vitess has been a core component of YouTube's database infrastructure since 2011, and has grown to encompass tens of thousands of MySQL nodes.
In addition to implementing the MySQL server and wire protocol, Vitess also provides the SQL parser for the database.
Dolt's use of Vitess
Dolt's fork of Vitess has headed in a different direction from the main project. In addition to adding support for DDL statements, stored procedures, and triggers (never a priority of the Vitess project), Dolt's fork prunes away the 90% of Vitess that isn't vital to Dolt's current roadmap. You can read details about this work in this blog post.