This is quite a common way of constructing expressions of astonishment and strong disagreement in most Germanic languages: using a subordinate clause and leaving out the main clause. Something like, “It shocks me [that…]” is then left unspoken, but pregnantly implied.
Google indicates that it works in German, too, though I’m not proficient enough myself to be quite certain (Afrikaans I tend to just ignore as being generally odd), and it certainly works in all the Scandinavian languages quite exactly as in Dutch: at du tør! (‘that you dare!’) or at du ikke skammer dig! (‘that you are not ashamed of yourself!’) in Danish should be almost directly intelligible to you.
However, as is often the case with these ‘pan-Germanic’ patterns and phenomena, English, in her insulation and Sinatricity1, has decided here that pan-Germanicisms are no good and got rid of it along the way—so the construction does not normally work in English, though there are still one or two remnants lying around (like Colin’s example, “That I should live to see this …” in the comment below, or something like, “That I never thought of this before …”). The semantic implication of the pan-Germanic pattern (that the thing mentioned not only astonishes, but also appals, you) is gone, though: only the element of unexpectedness is preserved.
A native English speaker would instead say, “How dare you?” and “You should be ashamed of yourself!” in the two cases given above. The subtle distinction the other Germanic languages are able to make between the two constructions cannot be expressed in English.
1 By which I mean that like Frank Sinatra, English must always do everything my way.