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/app.ts

@@ -7,6 +7,8 @@ import Fastify, { type FastifyInstance } from 'fastify';
 import { env } from './config/env.js';
 import { dbRoutes } from './modules/db/db.routes.js';
 import { healthRoutes } from './modules/health/health.routes.js';
+import { meRoutes } from './modules/me/me.routes.js';
+import { authPlugin } from './plugins/auth.js';
 import { prismaPlugin } from './plugins/prisma.js';
 
 export async function buildApp(): Promise<FastifyInstance> {
@@ -36,9 +38,11 @@ export async function buildApp(): Promise<FastifyInstance> {
   });
   await app.register(sensible);
   await app.register(prismaPlugin);
+  await app.register(authPlugin);
 
   await app.register(healthRoutes);
   await app.register(dbRoutes);
+  await app.register(meRoutes);
 
   return app;
 }