Depends on what this cookie is. If you're using curl to (say) log someone into Facebook by remote, you will not be able to set a facebook specific cookie on the client browser and have it be useable by Facebook. Remember, cookies are always set for their originating domain only. You could use curl to do the Facebook login sequence, extract the Facebook session cookie, and then send that cookie name/value pair to the client.
But as far as the client is concerned, that cookie came from your server's "example.com", not "facebook.com", so when the client then goes on to Facebook, they're not logged in, because the client has cookie that originated from facebook.com.
Even if it's not a login cookie and it's just a session ID, that session will not be properly useable by anything but code running on your server on behalf of the user, again because of this originating server security policy.