Silex / Lumen

They both provide minimum bootstrap.

Reading your comment about millions of records - has nothing to do with your framework and has all to do with SQL queries and how you code.

People who bash Lumen/Laravel + Symfony/Silex because its PHP and hop into NodeJS because of raw performance doesn't understand the architecture of scalability.

• Both provides json output which is what you need as an API (try not to give back XML, as i will go Liam Nielson on you)
• Request objects to ease up the code
• able to bootstrap various ORMs to your framework (yes lumen is compatible with eloquen since the core is the same - however you can bootstrap silex with eloquent since illimuminate /eloquent can be part of your composer)
• LTS both provides LTS support

I'm partial to Lumen because of pure community helpers - however if you try to do anything advance you might be stuck in a bind as i have faced. There are more out of the box packages compatible with Silex than Lumen/Laravel - however the community is getting more and more support to dev out compatiable drivers/packages. Watch out for documentation also - a lot of laravel base operations cannot be done in Lumen because it is a minimal framework - the minute you creep out of scope - you will want to ponder if it is not best to move to the big brother laravel in the end.

Lumen!

I hear that Slim is nice to make an API REST

I builded a api-platform.com based in Symfony Flex, and it is the best I have ever seen