This is for creating AWS API gateway using multiple stages along with lambda integration

aws-apigateway-lambda-with-multiple-stages.yaml

AWSTemplateFormatVersion: '2010-09-09'

Resources:
  LambdaApiGatewayRestApi:
    Type: AWS::ApiGateway::RestApi
    Properties:
      ApiKeySourceType: HEADER
      Description: An API Gateway with a Lambda Integration
      EndpointConfiguration:
        Types:
          - EDGE
      Name: lambda-api
  LambdaApiGatewayResource:
    Type: AWS::ApiGateway::Resource
    Properties:
      ParentId: !GetAtt LambdaApiGatewayRestApi.RootResourceId
      PathPart: 'lambda'
      RestApiId: !Ref LambdaApiGatewayRestApi
  LambdaApiGatewayMethod:
    Type: AWS::ApiGateway::Method
    Properties:
      ApiKeyRequired: false
      AuthorizationType: NONE
      HttpMethod: POST
      Integration:
        ConnectionType: INTERNET
        Credentials: !GetAtt LambdaApiGatewayIamRole.Arn
        IntegrationHttpMethod: POST
        PassthroughBehavior: WHEN_NO_MATCH
        TimeoutInMillis: 29000
        Type: AWS_PROXY
        Uri: !Sub 'arn:aws:apigateway:${AWS::Region}:lambda:path/2015-03-31/functions/arn:aws:lambda:us-east-1:{ACCOUNT_ID}:function:list_s3_files/invocations'
      OperationName: 'lambda'
      ResourceId: !Ref LambdaApiGatewayResource
      RestApiId: !Ref LambdaApiGatewayRestApi
  LambdaApiGatewayModel:
    Type: AWS::ApiGateway::Model
    Properties:
      ContentType: 'application/json'
      RestApiId: !Ref LambdaApiGatewayRestApi
      Schema: {}
  LambdaApiGatewayStage:
    Type: AWS::ApiGateway::Stage
    Properties:
      DeploymentId: !Ref LambdaApiGatewayDeployment
      Description: Lambda API Stage Stage
      RestApiId: !Ref LambdaApiGatewayRestApi
      StageName: 'Stage'
  LambdaApiGatewayProd:
    Type: AWS::ApiGateway::Stage
    Properties:
      DeploymentId: !Ref LambdaApiGatewayDeployment
      Description: Lambda API Stage Prod
      RestApiId: !Ref LambdaApiGatewayRestApi
      StageName: 'Prod'
  LambdaApiGatewayDeployment:
    Type: AWS::ApiGateway::Deployment
    DependsOn: LambdaApiGatewayMethod
    Properties:
      Description: Lambda API Deployment
      RestApiId: !Ref LambdaApiGatewayRestApi 
  LambdaApiGatewayIamRole:
    Type: AWS::IAM::Role
    Properties:
      AssumeRolePolicyDocument:
        Version: '2012-10-17'
        Statement:
          - Sid: ''
            Effect: 'Allow'
            Principal:
              Service:
                - 'apigateway.amazonaws.com'
            Action:
              - 'sts:AssumeRole'
      Path: '/'
      Policies:
        - PolicyName: LambdaAccess
          PolicyDocument:
            Version: '2012-10-17'
            Statement:
              - Effect: 'Allow'
                Action: 'lambda:*'
                Resource: 'arn:aws:lambda:us-east-1:{ACCOUNT_ID}:function:list_s3_files'
  
#{ACCOUNT_ID} - you can replace with your account id of aws account