Our soil is Bolar Clay Loam by USGS definition, 7.4 pH. It's as hard as a rock when it's dry. Being type A, I used a brute force solution to kick start our new garden.
Using a rented excavator, I made a little rain pond out of an unused area of our property. where we robbed soil. We used a rented dump trailer to move about 40 tons of that Bolar Clay, almost rock-free, and spread it to a depth of 6 to 10 inches on our garden site. Fortunately, weather was on our side because we had good rain a week or so ago. Were that clay hard, the big move would have been problematic to say the least.
Then I spread 15 tons of granite sand on top, with Disper-Sul Pastilles at a rate of 43 lbs per 1000 sqft, and roto-tilled the whole kit and caboodle up. We're working dairy manure into the top now as we plant. We'll continue with more gradual soil amendments over the years. We're also doing the cardboard and biomass method you mentioned.
It was a thousand bucks for machine rentals and took several days with five of us working. Two of our kids are still in homeschool so could only give us an hour or two here or there. All that cost and time may be a limiting factor for others. But heck, just a bag or two of sand from Lowes and you can get started on small areas. If you have a river nearby, river sand (sterilized) is even better.
Adding sand to clay is a long term partial solution. The sand won't go away. But you need organic matter in there too...and that must be replenished. Fortunately you don't have to go buy organic matter, you can generate it. Assuming you're here in the forum because you have chickens, put that manure to use as well.
Finally, the $30 you pay to have your state university extension test your soil is worth every penny. University extensions literally exist to help citizens use and improve their soil. Some claim that they have a heavy bias towards harsh chemical treatments. We have not found that to be true with Texas A&M Agri-Life. They discussed manures and renewable improvements.
Hope that helps....
