> First and foremost, you need setup serverless Node.js application. Based on the link to follow and configure as below:

# Welcome to Serverless!
#
# This file is the main config file for your service.
# It's very minimal at this point and uses default values.
# You can always add more config options for more control.
# We've included some commented out config examples here.
# Just uncomment any of them to get that config option.
#
# For full config options, check the docs:
#    docs.serverless.com
#
# Happy Coding!

service: MongoService
# app and org for use with dashboard.serverless.com
#app: your-app-name
#org: your-org-name

# You can pin your service to only deploy with a specific Serverless version
# Check out our docs for more details
frameworkVersion: '3'

provider:
  name: aws
  apiName: MongoService
  runtime: nodejs14.x
  region: ap-southeast-1

functions:
  hello:
    handler: handler.index
    name: getConnection
    description: Get PowerBI Connection from Amazon RDS
    events: 
      - http: 
          path: connection 
          method: post

Note: The key apiName allow us to change name of API gateway in serverless application.

> Run serverless deploy to deploy API Gateway and serverless to AWS server.

serverless deploy

Deploying MongoService to stage dev (ap-southeast-1)

✔ Service deployed to stack MongoService-dev (53s)

endpoint: POST - https://qgw9zym1o5.execute-api.ap-southeast-1.amazonaws.com/dev/connection
functions:
  hello: getConnection (5.9 MB)

Need a better logging experience than CloudWatch? Try our Dev Mode in console: run "serverless --console"

> Look at API Gateway dashboard

Note: The endpoint will target lambda function named getConnection(). If you open getConnection in Lambda and see the similarity.

> Here is postman for testing endpoint: https://qgw9zym1o5.execute-api.ap-southeast-1.amazonaws.com/dev/connection

> To detect what happens in lambda function, we can open CloudWatch for tracking it. So let move to monitor > View logs in CloudWatch.

For instance, here is your issue as below:

{
    "errorType": "Runtime.HandlerNotFound",
    "errorMessage": "handler.index is undefined or not exported",
    "stack": [
        "Runtime.HandlerNotFound: handler.index is undefined or not exported",
        "    at Object.module.exports.load (/var/runtime/UserFunction.js:304:11)",
        "    at Object.<anonymous> (/var/runtime/index.js:43:34)",
        "    at Module._compile (internal/modules/cjs/loader.js:1085:14)",
        "    at Object.Module._extensions..js (internal/modules/cjs/loader.js:1114:10)",
        "    at Module.load (internal/modules/cjs/loader.js:950:32)",
        "    at Function.Module._load (internal/modules/cjs/loader.js:790:12)",
        "    at Function.executeUserEntryPoint [as runMain] (internal/modules/run_main.js:75:12)",
        "    at internal/main/run_main_module.js:17:47"
    ]
}

> Here, the handler.js should change into index.js and need to exclude some files or folder that is unnecessary upon deployment.

# Welcome to Serverless!
#
# This file is the main config file for your service.
# It's very minimal at this point and uses default values.
# You can always add more config options for more control.
# We've included some commented out config examples here.
# Just uncomment any of them to get that config option.
#
# For full config options, check the docs:
#    docs.serverless.com
#
# Happy Coding!


service: MongoService
# app and org for use with dashboard.serverless.com
#app: your-app-name
#org: your-org-name


# You can pin your service to only deploy with a specific Serverless version
# Check out our docs for more details
frameworkVersion: '3'

provider:
  name: aws
  apiName: MongoService
  runtime: nodejs14.x
  region: ap-southeast-1

layers:
  database:
    package:
      artifact: mongodb.zip
    name: database
    description: contain libs for connecting mongo db
    compatibleRuntimes:
      - nodejs14.x
    retain: true
    allowedAccounts:
      - '*' # ALL accounts!

functions:
  hello:
    handler: index.handler
    name: getConnection
    description: Get PowerBI Connection from Amazon RDS
    layers:
      - arn:aws:lambda:ap-southeast-1:181156772356:layer:mongodb:3
    events: 
      - http: 
          path: connection 
          method: post

package:
  exclude:
    - node_modules/**
    - package.json
    - package-lock.json
    - database.zip
    - run.ts
  include:
    - index.js

Note: You need to run layers first and copy arn:aws:lambda:ap-southeast-1:181156772356:layer:database:3 after generating. To set up for function, we need create layers and assign value again.

> Go to Postman and run again.

> You can download source code from link here. If you got problem then you need to go to CloudFormation > Stacks. You can delete all stacks to ensure serverless deploy work properly.