Converted example to TypeScript to match Contribution docs.
- Set cookies using `NextResponse` instead of the `Set-Cookie` header
## Documentation / Examples
- [X] Make sure the linting passes by running `pnpm build && pnpm lint`
- [X] The "examples guidelines" are followed from [our contributing doc](https://github.com/vercel/next.js/blob/canary/contributing/examples/adding-examples.md)
