Sheetmulching without soil?

Since some months I am sheetmulching (= lasagna gardening, or however you will call that). It is not by choice but our "soil" dictates that.
As it is winter and everything grows very slow I cannot yet say weather it works.

I put a generous layer of cardboard on the bottom, then I put down wooden mulch about 5-10 cm. On the top of that 10cm of finely cut up branches, then 10cm of soil, 10 mushroom compost, and at least 10 horse manure.
The soil is difficult to come by, I have not possibility to dig it up in the garden and either I buy topsoil at $40 a ton (I have to buy 12 tons at the time) or I get something which is full of rubbish like tiles concrete etc.
Leaving the soil out? How can I replace the soil? And what about minerals?

I use a similar system, but with a different order; I dig a hole or trench, fill with good soil and plant into that, then spread out 2cm layer of compost over everything including weeds, cardboard over everywhere not planted, 2cm compost over cardboard only, wood mulch and then branches on top(I use straw as well). The idea is to give the plants a little area to grow and then feed the mushrooms and bacteria to breakdown the woody material and create carbon rich humus, the start of healthy soil.
The problem you may be experiencing is one of aeration. Open up your lasagna and look around in the compost for ashy looking areas. This is an indicator of anaerobic compost, which is actually bad for your plants.

I did some sheet mulching and planting without soil this year, since I just moved to a new place and the soil is rock hard clay and silt.
I thought my garden wouldn't grow much, but it seems to be doing OK. Not gangbusters, but OK.
I made raised beds and filled them with a mix of about 2/3 leaf mulch (from last year's leaf piles, both mine and all I could filch from my neighbors) and rotten straw I was given, and 1/3 aged horse poop. I had thought just to layer things, but I only had the two layers, so I mixed them up. I had put down torn open paper bags as a weed barrier under the mulch and poop, rather than cardboard, since that is what I had. Then the leaf mulch/rotten straw, then the horse manure. I lightly mixed up the leaf/straw with the horse manure, and let it sit for a week before planting in it. Oh, and I mixed in a bunch of wood ashes too, since I thought the leaf mulch might be a little acidic. I sprinkled it on fairly generously before mixing it all up.

Everything was slow to start, but I now have peas, mizuna, spinach, strawberries, and, surprisingly, carrots all doing pretty well. Radishes were great, and the biggest success is the onions. Usually my onions never get very big, but the ones that I just tucked into the poopy mulch a month or so ago are now hanging out mostly above the surface and are really big. By the time they are ready to harvest they will be big like grocery store onions, which will be a first for me.

The added organic matter (poop, straw, leaves, ashes) had started out being about 6 or 7 inches thick when I first put it in the beds, but now it is a lot less thick than that. But the soil underneath is not so rock hard, so I'm hopeful that after a year or two the garden beds will be really nice.