Ensure IntegrationHttpMethod is set to POST even if you want to use a GET (or whatever)

e.g.

  rootMethod:
    Type: AWS::ApiGateway::Method
    Properties:
      AuthorizationType: NONE
      HttpMethod: GET
      Integration:
        IntegrationHttpMethod: POST
        Type: AWS_PROXY
        Uri: !Sub "arn:aws:apigateway:${AWS::Region}:lambda:path/2015-03-31/functions/${lambdaArn}/invocations"
      ResourceId: !GetAtt apiGateway.RootResourceId
      RestApiId: !Ref apiGateway

See this thread: https://forums.aws.amazon.com/thread.jspa?messageID=745275

When attempting to set up an API Gateway with a single endpoint that used a Lambda we continually got the following error message when attempting to use GET methods:

<AccessDeniedException>
  <Message>Unable to determine service/operation name to be authorized</Message>
</AccessDeniedException>

This was apparently due to the fact that regardless of the actual HttpMethod we set, IntegrationHttpMethod needs to be set to POST as per this piece of AWS documentation:

https://docs.amazonaws.cn/en_us/apigateway/latest/developerguide/set-up-lambda-custom-integrations.html

For Lambda integrations, you must use the HTTP method of POST for the integration request, according to the specification of the Lambda service action for function invocations. The uri parameter is the ARN of the function-invoking action.