Add TanStack Start server code example for chat API

The Next.js-only code was confusing for a beginner unfamiliar with Tanstack syntax.

Author: harlley

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";