Hi I don't think there is any need for the second " that " in your first sentence. That's a nice hat you're wearing. Your second sentence sounds odd to my non-native ears.
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Debpriya Dewhy can't we write "That that you do is illegal"?You can write it, and it is grammatical, but it is almost universally regarded as inferior in style. It just sounds strange when you say the same word twice in a row, so such patterns are usually avoided.