The thing that I have done is to catch the thief.
我感觉这个句子是错的.正确地表述方式为：
The thing that I have to do is to catch the thief.我所要做的就是抓住小偷.或
The thing I have done is that I (have) caught the thief.我所做的就是抓住了小偷.
[I主语] [haven't made up谓语] [my mind宾语] [whether to go there or not宾语补语].