docs: clarify setting and reading cookies from Route Handlers (vercel…

…#59915)

Replaces vercel#57052

docs/02-app/01-building-your-application/01-routing/10-route-handlers.mdx

@@ -229,9 +229,9 @@ Route Handlers can be used with dynamic functions from Next.js, like [`cookies`]
 
 #### Cookies
 
-You can read cookies with [`cookies`](/docs/app/api-reference/functions/cookies) from `next/headers`. This server function can be called directly in a Route Handler, or nested inside of another function.
+You can read or set cookies with [`cookies`](/docs/app/api-reference/functions/cookies) from `next/headers`. This server function can be called directly in a Route Handler, or nested inside of another function.
 
-This `cookies` instance is read-only. To set cookies, you need to return a new `Response` using the [`Set-Cookie`](https://developer.mozilla.org/docs/Web/HTTP/Headers/Set-Cookie) header.
+Alternatively, you can return a new `Response` using the [`Set-Cookie`](https://developer.mozilla.org/docs/Web/HTTP/Headers/Set-Cookie) header.
 
 ```ts filename="app/api/route.ts" switcher
 import { cookies } from 'next/headers'
@@ -261,7 +261,7 @@ export async function GET(request) {
 }
 ```
 
-Alternatively, you can use abstractions on top of the underlying Web APIs to read cookies ([`NextRequest`](/docs/app/api-reference/functions/next-request)):
+You can also use the underlying Web APIs to read cookies from the request ([`NextRequest`](/docs/app/api-reference/functions/next-request)):
 
 ```ts filename="app/api/route.ts" switcher
 import { type NextRequest } from 'next/server'
@@ -311,7 +311,7 @@ export async function GET(request) {
 }
 ```
 
-Alternatively, you can use abstractions on top of the underlying Web APIs to read headers ([`NextRequest`](/docs/app/api-reference/functions/next-request)):
+You can also use the underlying Web APIs to read headers from the request ([`NextRequest`](/docs/app/api-reference/functions/next-request)):
 
 ```ts filename="app/api/route.ts" switcher
 import { type NextRequest } from 'next/server'