Advice Needed – what are my options for gardening next to huge cottonwoods?

We moved into a new house less than a year ago and I’ve been very eager to set up my own garden in the yard, but our entire yard is taken over by the roots of our neighbour’s cottonwood. The spot where I’d like to plant is where a 40 year old crab apple tree that we cut down last fall was (red circle). The apple tree was 15-20’ tall, flowered heavily and produced way too much worm infested fruit. We didn’t do anything to the roots and simply chopped it down to ground level. So it’s mostly apple tree roots directly below the area I’d like to use for gardening. Is my only option to place solid bottomed raised beds? FWIW I’m zone 3B/4A and the photo is from October 1st last year.