【1】当“way”表示“方法，方式”时，后面接 of doing 或者to do

He's trying to figure out a way to solve the problem.

他正在努力想办法解决这个问题。

We have found a way of safeguarding our money.

我们已有了保护钱财的办法。

【2】当“way”表示“道路,路径”时，后面的to是介词，意为“通向......之路”后面接way to sth 或者way to doing sth

Self-help is the best way to success.  自助乃最佳成功之道。

By following these four steps, you’ll be on your way to getting credit for your knowledge and accomplishments.

通过以下四个步骤，你可以为你的知识和成就获得相应的学分。

【3】他快要成为工程师了。有以下几种翻译

He is on the way to becoming an engineer.

He is about to become an engineer.

He is on the point of becoming an engineer.

He is on the verge of becoming an engineer.

He is going to be an engineer very soon.

         be on the way to 中的 to 通常是个介词，后接名词或 v-ing 形式，介词短语在句中多作表语，意思是“在去......的路上”。

        ......

        但偶尔也可见到 be on the way to  中的 to 作不定式符号的句子。例如：He is on his way to see a film. 他正在去看电影的路上。/ He is on the way to succeed. 他在成功的道路上。

 （以上摘自《薄冰英语惯用法词典》第1297页）

   所以，on the way to do 的说法是正确的，只是不普通而已。