Cloudformation Cognito - how to setup App Client Settings, Domain, and Federated Identities via SAM template

I have created two CloudFormation custom resources to apply Cognito app client settings and domain name. With these resources, you can have a script like this:

UserPoolTestClient:
  Type: 'AWS::Cognito::UserPoolClient'
  Properties:
    ClientName: UserPoolTestClient
    GenerateSecret: true
    UserPoolId: !Ref UserPoolTest
UserPoolTestClientSettings:
  Type: 'Custom::CognitoUserPoolClientSettings'
  Properties:
    ServiceToken: !GetAtt CloudFormationCognitoUserPoolClientSettings.Arn
    UserPoolId: !Ref UserPoolTest
    UserPoolClientId: !Ref UserPoolTestClient
    SupportedIdentityProviders:
      - COGNITO
    CallbackURL: 'https://www.amazon.com'
    LogoutURL: 'https://www.google.com'
    AllowedOAuthFlowsUserPoolClient: true
    AllowedOAuthFlows:
      - code
    AllowedOAuthScopes:
      - openid
UserPoolTestDomain:
  Type: 'Custom::CognitoUserPoolDomain'
  Properties:
    ServiceToken: !GetAtt CloudFormationCognitoUserPoolDomain.Arn
    UserPoolId: !Ref UserPoolTest
    Domain: 'userpool-test-01'

The complete code is here.

Looks like there is no way to provide App integration -> Domain name and Federation -> Identity providers via CloudFormation.

I found only reference for User Pool Client (General settings -> App clients) but it will not configure App integration -> App client settings.

If you need to automate process of providing Domain name, Identity providers and App client settings for user pool, you can do that by creating custom script (AWS CLI) or Lambda (AWS SDK) which should be performed after stack deployment.

UPDATE

Check out excellent example (answer below) that shows usage of CloudFormation Custom Resources with Lambda.

CloudFormation has added the resource AWS::Cognito::UserPoolDomain to manage the User Pool Domain:

Type: AWS::Cognito::UserPoolDomain
Properties: 
  CustomDomainConfig: 
     CertificateArn: !Ref CertificateArn
  Domain: "your.custom.domain.com"
  UserPoolId: !Ref UserPool

In addition, there has been added configuration to the AWS::Cognito::UserPoolClient:

Type: AWS::Cognito::UserPoolClient
Properties: 
  AllowedOAuthFlows: 
    - String
  AllowedOAuthFlowsUserPoolClient: Boolean
  AllowedOAuthScopes: 
    - String
  AnalyticsConfiguration: 
    AnalyticsConfiguration
  CallbackURLs: 
    - String
  ClientName: String
  DefaultRedirectURI: String
  ExplicitAuthFlows: 
    - String
  GenerateSecret: Boolean
  LogoutURLs: 
    - String
  ReadAttributes: 
    - String
  RefreshTokenValidity: Integer
  SupportedIdentityProviders: 
    - String
  UserPoolId: String
  WriteAttributes: 
    - String