Serverless Cognito - Sign Up with Username, Password, and Email - amazon-web-services

Question

I am having trouble implementing Cognito via Serverless. The AWS Cognito documentation is poor. What I want to do is:
Allow user to sign up with a unique username (coolprogrammer69), a unique email, and password
After sign up, allow user to modify username (coolprogrammer69 -> awesomeprogrammer69) and email as long as it is unique
This is a typical auth flow for a forum website like Stackoverflow; however, I cannot find the right combination of attributes to make this a reality. Can you please help me figure out what I am doing wrong?
service: my-cool-infrastructure
package:
individually: true
provider:
name: aws
runtime: nodejs14.x
environment:
user_pool_id: { Ref: UserPool }
client_id: { Ref: UserClient }
iamRoleStatements:
- Effect: Allow
Action:
- cognito-idp:AdminInitiateAuth
- cognito-idp:AdminCreateUser
- cognito-idp:AdminSetUserPassword
Resource: "*"
functions:
login:
handler: user/login.handler
events:
- http:
path: user/login
method: post
cors: true
register:
handler: user/register.handler
events:
- http:
path: user/register
method: post
cors: true
privateAPI:
handler: user/private.handler
events:
- http:
path: user/private
method: post
cors: true
authorizer:
name: PrivateAuthorizer
type: COGNITO_USER_POOLS
arn:
Fn::GetAtt:
- UserPool
- Arn
claims:
- email
resources:
Resources:
UserPool:
Type: AWS::Cognito::UserPool
Properties:
UserPoolName: my-cognito-pool
Schema:
-
Name: email
Mutable: true
Required: true
-
Name: preferred_username
Mutable: true
Required: true
Policies:
PasswordPolicy:
MinimumLength: 8
RequireNumbers: true
RequireSymbols: true
UserClient:
Type: AWS::Cognito::UserPoolClient
Properties:
ClientName: user-pool-ui
GenerateSecret: false
UserPoolId: { Ref: UserPool }
AccessTokenValidity: 5
IdTokenValidity: 5
ExplicitAuthFlows:
- "ADMIN_NO_SRP_AUTH"