Preheat the oven to 400ºF. Cut squash in half lengthwise and scoop out the seeds. Place squash cut side down in a baking dish and add 1/4" in of water. Roast for 40 - 50 minutes, until a knife can easily penetrate the skin.
Remove from oven, flip over and let sit until cool enough to handle. Scoop out the flesh with a fork into a clean, dry dishtowel. Squeeze as much water as you can out of the squash. Transfer to a mixing bowl.
Add remaining ingredients to a bowl and fold to combine. The mixture should start to resemble a soft dough.
Heat oil in a skillet over medium-high heat. Add cakes and cook until golden on one side, 2 - 3 minutes. Flip and cook for another 1 - 2 minutes. Repeat with remaining cakes.
Serve immediately with a side of salsa, hummus or tomato sauce.