Preheat an oven to 400 degrees F (200 degrees C). Prepare a baking sheet with cooking spray.
Toss together the potatoes, carrot, parsnip, and squash in a large bowl until evenly mixed. Tilting the bowl away from you, collect all the vegetables in your hands and squeeze them to separate the juice. Move the vegetables into a separate bowl and mix in the cinnamon.
After 5 minutes or so, slowly pour out the juice from the first bowl – the bottom of the bowl should be coated with starch. Combine the starch with egg and flour, using a fork to stir it together. Then take the starch mixture in your hands and massage it through the shredded vegetables. Separate the mix into little balls, about 1 inch thick, and flatten onto the prepared baking sheet.
Bake the patties in the oven until lightly browned around the edges, about 20 minutes; flip and bake until golden brown, about 5 minutes more.