Is there a way to respond to 你做什么工作？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.