How do I attach a flat/gable verandah to an existing house?
I'd build out from your existing roof, in line with the peak on the new verandah. (Call this a dormer.) Fill in the gable end of the new dormer on the existing roof (just as in the picture you provided), and you won't have problems with leakage in the door area. (Nor would you need a gutter.) The only faintly tricky part of this is that you'd need to tie the new roof into your existing tiles, but that's what roofers do all the time.
If this isn't clear, ping back and I'll post a picture.

Edit one: a picture, so we're on the same page... (and some massaging of the text above in hopes of clarity).

- red is your existing roof
- green is the wall of the existing house
- blue is the top of the dormer that would be added onto your roof
- yellow is the gable end of the new dormer
- the wireframe is the rest of your new verandah

So, filling in the yellow bit is very straightforward, because you've got structure behind in the new blue dormer (that I've suggested you build). Because you have that new dormer, no rain will end up near your door. (You will need to do a little bit of work making sure there's enough slope to drain water off the flatter bits of the verandah, though.)

Clear as mud? If not, ping back.