HospitalController.php

<?php
...
 class HospitalCnotroller{
  public function index(Request $request){
    $hospitas = Hospital::all()->pluck('nome','id');
    $dados = Servidor::where('hospital',$request->hospital)->paginate();
    return view('servidores',['dados'=>$dados,'hospital'=>$hospital,'hospitais'=>$hospitais]);
  }
}

servidores.blade.php

<form>
<select>
@foreach($ospitais as $hosp)
<option   value="{{$hosp->id}}">{{$hosp->name}}</option>
@endforeach
</select>
</form>
@foreach($dados as $dado)
<tr><td>{{$dado->nome}}</td><td>{{$dado->sobrenome}}</td></tr>
@endforeach
@dados->append(['hospital'=>$hospital])->links()