Is there a way to respond to 你做什么工作？starting with 我做...？ For many questions in Chinese, it's best to respond by repeating the verb.  Here's a few I just made up:

Q: 你是哪儿国人？ A: 我是中国人。
Q: 刘老师找什么？ A: 刘老师找她的老公。
Q: 她有老公吗？ A: 对，她有老公。
Q: 小王怕什么？ A: 小王怕狗。

And I can think of many others (你想吃什么？ 你在看什么？你在学什么？你的牙疼吗?).  But this one breaks the rules:

Q: 你做什么工作？
A: 我是医生

Or we might respond 我是老师 or 我是校长, or 我是职员, etc., depending on our job.
Question: Is there a way to respond to 你做什么工作？ starting with 我做...？
Perhaps there's something inherently wrong with answering the question starting with 我做....
Because 医生 is what you are, not what you do.

You can respond with 我教英语, 我卖面包, 我开出租车, if you want.

In most cases, 我是XX sounds more natural if your job has a simple and direct name, such as 医生, 教师, 司机, 厨师, etc. The other pattern is more likely to be used when you want to specifically describe what you do.