Roofing over basement stairs for terrace extension. How to execute?

Hello everyone,

I would like to enlarge my terrace and for that, part of the cellar staircase would need to be covered.
With the terrace extension, about 30cm of height would also need to be compensated, which means the covering must be well load-bearing (structural?).

Do you have any ideas on how this could be realized?
I was maybe thinking of formwork from below and then pouring a concrete slab in 2-3 steps?!
And somehow working with lintels and steel beams?! Especially the fastening to the house with 18cm [WDVS] I find difficult.

How is this best done?
I tried to illustrate it on 3 pictures.

What kind of surface does the current terrace have. Do you want a completely new one, or just patch it up. For a completely new one, I would use decking boards and build a beam structure underneath in the area of the stairs. When adding on with a panel, there will probably be a constant crack at the transition.

Is there a plan of the stairs / basement? Ceiling height?

Concrete walls?

I see the problem with a development that is opaque in that sooner or later you have mud and standing water in the lower area of the cellar stairs. No light or air then gets in to dry everything out.

So the basement wall is lean concrete. The current terrace has gravel+concrete slabs. I plan to completely redo the terrace. However, I would like to keep the existing gravel base due to the effort and patch it up or fill around it. The basement staircase itself has a water drainage, so the water flows away. In addition, even less water reaches there if it is covered. Therefore, I wouldn't be too worried about that now. Do the wooden beams also withstand the weather? I mean, I would still have to fill up 30-40cm above them. That's quite a bit of weight.