Came late to this argument about taking 2 QB's w/our first 2 picks. Of course, you don't do that. With the holes we have, the still fantastic talent that is going to be available at #4, pretty much a game-changer from Day 1 (Minkah or Chubb, throw in Barkley if you want, who gives a shit (even though I don't want him at 4)), you don't take 2 QB's when you already have a young QB you just fucking drafted on your roster. Sign a vet-bridge and if the bridge is good enough to run ship for a year or 2, let the young guy learn. If he's literally there to mentor the young QB (see: Rosen) then when Rosen is ready to go, throw him in there *cough*McCown*cough*
Also, what wasn't brought up - you are absolutely doing a huge disservice to both of those QB's. You can joke all you want about the Browns being the place where college QB's go to die, but you are robbing one of those QB's the chance to have a successful career in the NFL from the get-go b/c one of them would be sitting doing jack shit. Not just actual physical play, but mentally as well.
If you want to draft 2 QB's, which I just feel is so dumb and a wasted pick - wait until the mid-later rounds and select someone like Lauletta, Falk, or Ferguson. They aren't coming into the league with expectations to start right away or even a year or 2 down the line. So bring one of them in if you really want to.