It is not possible to invoke a Lambda function asynchronously with a proxy integration. You can do this with Lambda non-proxy integration.
Set up asynchronous invocation of the backend Lambda function - https://docs.aws.amazon.com/apigateway/latest/developerguide/set-up-lambda-integration-async.html
Another workaround is to use another function that is invoked synchronously by the API Gateway API and have that function invoke the function that is part of the Spring boo application asynchronously using the Lambda Invoke API or SDK equivalent.
https://docs.aws.amazon.com/lambda/latest/dg/API_Invoke.html
For instance, you want Function A to be invoked asynchronously. To achieve this you would have the API Gateway API invoke Function B synchronously and have Function B invoke Function A asynchronously with the Invoke API.
Step Functions also cant be used as a workaround to invoke the REST APIs as this would not help. The Lambda functions are still invoked by the API Gateway API so you would still run into the limitation of a proxy integrations not being able to invoke Lambda asynchronously. You then decided that ECS would better suit your case as Lambda is not a good fit.
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