We will use proxy integration for our API, in order to pass the request to the Lambda without any changes. We will use the greedy path param {proxy+} under the root resource , in order to catch all of the requests to its sub-resources. We will also add the ANY HTTP method over the greedy path to match for any type.
The requests to the root resource, are still not matched. If we need to match for , as well, we should define and configure an ANY method over it.