feat(api): add Better Auth plugin and protected GET /me endpoint

- authPlugin runs preHandler that turns Fastify request headers into a Headers
  object, calls auth.api.getSession, and decorates request.user / request.session
- requireAuth() preHandler short-circuits with 401 when not signed in
- New module modules/me with GET /me returning the authenticated user/session
- env validates BETTER_AUTH_SECRET (≥32) and BETTER_AUTH_URL — must match web
- Restored .js extensions in shared packages so NodeNext-resolution consumers
  (api) typecheck cleanly; Next webpack now uses extensionAlias to map .js → .ts
- Re-enabled NodeNext for packages/auth and packages/db tsconfigs

Verified end-to-end:
- POST /api/auth/sign-in/email on web returns session cookie
- GET /me on api with the cookie returns 200 + user/session
- GET /me without the cookie returns 401

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

apps/api/src/modules/me/me.routes.ts

@@ -0,0 +1,10 @@
+import type { FastifyPluginAsync } from 'fastify';
+
+import { requireAuth } from '../../plugins/auth.js';
+
+export const meRoutes: FastifyPluginAsync = async (fastify) => {
+  fastify.get('/me', { preHandler: requireAuth() }, async (request) => ({
+    user: request.user,
+    session: request.session,
+  }));
+};