Add TanStack Start server code example for chat API

docs/getting-started/quick-start.md

@@ -19,9 +19,64 @@ yarn add @tanstack/ai @tanstack/ai-react @tanstack/ai-openai
 
 First, create an API route that handles chat requests. Here's a simplified example:
 
+### TanStack Start
+
+```typescript
+import { chat, toStreamResponse } from "@tanstack/ai";
+import { openai } from "@tanstack/ai-openai";
+import { createFileRoute } from "@tanstack/react-router";
+
+export const Route = createFileRoute("/api/chat")({
+  server: {
+    handlers: {
+      POST: async ({ request }) => {
+        // Check for API key
+        if (!process.env.OPENAI_API_KEY) {
+          return new Response(
+            JSON.stringify({
+              error: "OPENAI_API_KEY not configured",
+            }),
+            {
+              status: 500,
+              headers: { "Content-Type": "application/json" },
+            },
+          );
+        }
+
+        const { messages, conversationId } = await request.json();
+
+        try {
+          // Create a streaming chat response
+          const stream = chat({
+            adapter: openai(),
+            messages,
+            model: "gpt-4o",
+            conversationId,
+          });
+
+          // Convert stream to HTTP response
+          return toStreamResponse(stream);
+        } catch (error) {
+          return new Response(
+            JSON.stringify({
+              error:
+                error instanceof Error ? error.message : "An error occurred",
+            }),
+            {
+              status: 500,
+              headers: { "Content-Type": "application/json" },
+            },
+          );
+        }
+      },
+    },
+  },
+});
+```
+
+### Next.js
+
 ```typescript
-// app/api/chat/route.ts (Next.js)
-// or src/routes/api/chat.ts (TanStack Start)
 import { chat, toStreamResponse } from "@tanstack/ai";
 import { openai } from "@tanstack/ai-openai";