search.js

const sample = [
  [
    [
      [
        [
          [
            {
              foo: [{ rate: 2.18 }],
            },
          ],
        ],
      ],
    ],
  ],
  {
    fieldCount: 0,
    affectedRows: 0,
    insertId: 0,
    serverStatus: 0,
    message: '',
    protocol141: true,
    changedRows: 0,
  },
];

const search = (object, target, defaultValue) =>
  Object.entries(object).reduce((stack, [key, value]) => {
    switch (true) {
      case key === target:
        return value;
      case !!Object.keys(value).length:
        return search(value, target, stack);
      default:
        return stack;
    }
  }, defaultValue);

console.log(search(sample, 'rate')); // 2.18
console.log(search(sample, 'protocol141')); // true
console.log(search(sample, 'hueBR', 'LOL')); // 'LOL'

Boa!