<?php
$cooking = [
  ["menu_id" => "10","menu_name" => "カレー","submenu_name" => "ライス","drink" => "烏龍茶"],
  ["menu_id" => "20","menu_name" => "パスタ","submenu_name" => "タバスコ","drink" => "ビール"],
  ["menu_id" => "30","menu_name" => "焼き肉","submenu_name" => "ソース","drink" => "ジンジャエール"]
];

//↓一番早い
$menu = array_column($cooking, 'menu_name', 'menu_id');

$menu = array_column($cooking, null, 'menu_id');

↓２番早い  配列操作を覚えるには丁度よい
foreach ($cooking as  $row) {
    $data[$row['menu_id']] = $row['menu_name'];
}



var_dump($menu);

#結果

array(3) {
  [10]=>
  string(9) "カレー"
  [20]=>
  string(9) "パスタ"
  [30]=>
  string(9) "焼き肉"
}