Failed to create API Gateway - amazon-web-services

Question

I'm trying to create this API Gateway (gist) with Authorizer, and ANY method.
I run into this error:
The following resource(s) failed to create: [BaseLambdaExecutionPolicy, ApiGatewayDeployment]
I've checked the parameters passed into this template from my other stacks and they're correct. I've checked this template and it's valid.
My template is modified from this template with "Runtime": "nodejs8.10".
This is the same stack (gist) which is created successfully using swagger 2. I just want to replace swagger 2 with AWS::ApiGateway::Method.
Update 6 Jun 2019:
I tried to create the whole nested stack using the working version of the API Gateway stack, then create another API Gateway with the template that doesn't work with the parameters I get from the nested stack, then I have this:
The REST API doesn't contain any methods (Service: AmazonApiGateway; Status Code: 400; Error Code: BadRequestException; Request ID: ID)
But I did specify the method in my template following AWS docs:
"GatewayMethod": {
"Type" : "AWS::ApiGateway::Method",
"DependsOn": ["LambdaRole", "ApiGateway"],
"Properties" : {
"ApiKeyRequired" : false,
"AuthorizationType" : "Cognito",
"HttpMethod" : "ANY",
"Integration" : {
"IntegrationHttpMethod" : "ANY",
"Type" : "AWS",
"Uri" : {
"Fn::Sub": "arn:aws:apigateway:${AWS::Region}:lambda:path/2015-03-31/functions/${LambdaFunction.Arn}/invocations"
}
},
"MethodResponses" : [{
"ResponseModels": {
"application/json": "Empty"
},
"StatusCode": 200
}],
"RequestModels" : {"application/json": "Empty"},
"ResourceId" : {
"Fn::GetAtt": ["ApiGateway", "RootResourceId"]
},
"RestApiId" : {
"Ref": "ApiGateway"
}
}
},

Answer 1

Thanks to #John's suggestion. I've tried to create the nested stack with the version that worked and pass in the parameters for the version that doesn't work.
The reason for that error is:
CloudFormation might try to create Deployment before it creates Method
from balaji's answer here.
So this is what I did:
"methodANY": {
"Type": "AWS::ApiGateway::Method",
"Properties": {
"AuthorizationType": "COGNITO_USER_POOLS",
...},
"ApiGatewayDeployment": {
"Type": "AWS::ApiGateway::Deployment",
"DependsOn": "methodANY",
...
I also found this article on cloudonaut.io by Michael Wittig helpful.