First to answer your question: I used a pick, a digging bar and the grubbing end of a pulaski to break up the ground. Hard work.
I have the same land. Ultra-compacted clay. I made some garden beds there doing the following:
I dug it up about 1 foot deep. Picked out rocks and the pure white clay chunks that might as well be rocks.
Mixed in wood pieces, horse manure and some biochar.
Planted a green manure cover crop and let it grow for 6 months.
Then I mounded it all up into beds, and covered all of it with 4 inches of wood chips. I also filled in the paths with wood chips to a depth of about 8 inches. After this the green manure crops popped out again.
I scraped off the wood chips and the green manure in the spring and mulched with some compost, then re-covered with the chips/green manure. Then I doused it with aerated compost tea and planted.
Results were pretty ok. The garden started great but ended up "self compacting" over the course of the season. Upon investigation the actual growing depth of the beds is more like 6 inches (they are almost 18" high) and under that is just nasty water-phobic clay. Tomatoes grew ok, Zucchini and Lemon Cucumber grew great. Peppers and Melons sucked. Pickling cucumbers are so bitter as to be inedible.
I will be incorporating about 400% more organic materials and biochar into as many beds as I can before next spring. I think I will be in business after that.
Anyways, food for thought I guess.