In a similar situation of prepping an overgrown pasture we employed a minimal till strategy just to break up that thick sod layer. (David the Good has an informative YouTube video on this)
We used a little tractor w/ potato plow to flip over the sod to kill it. Then we went at with hoes to break up the clumps (crop-fit!). Finally when everything was broken up we used a Hoss manual wheel hoe to further breakdown the now dead grass and mix in a small amount of manure and compost to make the beds fluffy and fertile for planting.
I was happy to see tons of squiggly worms during the process and we ended up with a respectable harvest the very first growing season.
The entire process took @ 1 month or so from breaking ground to planting.
When flipping beds the next time we just do a very light hoe to reincorporate new compost/manure and whatever garden refuse is left from the prior crop.
Happy Planting!