想定

クロスリージョンスタック(複数リージョンのスタックを連携)
NodejsFunctionを使っている（ts で書いたハンドラーをそのままデプロイできる construct）

結論

cdk.ts

#!/usr/bin/env node
import * as cdk from "aws-cdk-lib";
import "source-map-support/register";
import { LambdaEdgeStack } from "../lib/lambda-edge-stack";
import { SampleStack } from "../lib/sample-stack";

const app = new cdk.App();

const account = process.env.CDK_DEFAULT_ACCOUNT;

export const REGION = {
  VIRGINIA: "us-east-1",
  TOKYO: "ap-northeast-1",
};

const lambdaEdgeStack = new LambdaEdgeStack(app, LambdaEdgeStack.name, {
  env: { account, region: REGION.VIRGINIA },
  crossRegionReferences: true,
});

const sampleStack = new SampleStack(app, SampleStack.name, {
  env: { account, region: REGION.TOKYO },
  lambdaEdgeFunction: lambdaEdgeStack.function,
  crossRegionReferences: true,
});

sampleStack.addDependency(lambdaEdgeStack);

lambda-edge-stack.ts

import { Stack, StackProps } from "aws-cdk-lib";
import { CompositePrincipal, ManagedPolicy, Role, ServicePrincipal } from "aws-cdk-lib/aws-iam";
import { Runtime } from "aws-cdk-lib/aws-lambda";
import { NodejsFunction } from "aws-cdk-lib/aws-lambda-nodejs";
import { Construct } from "constructs";
import { join } from "path";

export class LambdaEdgeStack extends Stack {
  function: NodejsFunction;

  constructor(scope: Construct, id: string, props?: StackProps) {
    super(scope, id, props);

    const role = new Role(this, "LambdaEdgeExecutionRole", {
      assumedBy: new CompositePrincipal(
        new ServicePrincipal("lambda.amazonaws.com"),
        new ServicePrincipal("edgelambda.amazonaws.com") // デフォルトで生成されるロールはこれがないので追加
      ),
      managedPolicies: [ManagedPolicy.fromAwsManagedPolicyName("service-role/AWSLambdaBasicExecutionRole")],
    });

    this.function = new NodejsFunction(this, NodejsFunction.name, {
      entry: join(__dirname, "./handler.ts"),
      handler: "handler",
      runtime: Runtime.NODEJS_18_X,
      awsSdkConnectionReuse: false, // このプロパティを`false`にしないと、`LambdaEdge`は環境変数は使えないよエラーを出す
      role,
    });
  }
}

handler.ts

import { Callback, CloudFrontRequestEvent, Context } from "aws-lambda";

export const handler = async (event: CloudFrontRequestEvent, context: Context, callback: Callback) => {
  console.log("hi mom")
};

ポイント

lambda-edge-stack.ts で awsSdkConnectionReuse: false
cloudfront でのログ出力のためlambda-edge-stack.ts でロールにedgelambda.amazonaws.comを付与必須

1 を true にすると lambdaEdge では環境変数は使えないエラーが出ます。
2 はデフォルトのロールも生成されるのですが、それだと ServicePrincipal にlambda.amazonaws.comしかないので、edgelambda.amazonaws.comを追加しないと

cdk デプロイ時に

Resource handler returned message: "Invalid request provided: AWS::CloudFront::Distribution: The function execution role must be assumable with edgelambda.amazonaws.com as well as lambda.amazonaws.com principals. Update the IAM role and try again.

というエラーが出るのでそのため新しくロールを作成しています。

参考

https://docs.aws.amazon.com/ja_jp/AmazonCloudFront/latest/DeveloperGuide/lambda-edge-permissions.html#lambda-edge-permissions-function-execution