Trim ends from the squash. Peel the squash and cut it in half lengthwise. Scoop out the seeds. Cut each half into quarters, then finely slice each pieces into 1/8-inch thick slices.
Heat the oil over a medium flame in a medium, heavy-bottomed saucepan. Add the onion and cook, stirring until soft, about 3 minutes. Add the squash, season with salt, and add water just to cover.
Bring the squash to a simmer and cook until it easily breaks apart, about 20 minutes.
Stir in the rice and 1 cup broth. Bring to a rolling boil and cook, stirring frequently and adding more cups of broth as needed, for 8 minutes. The rice should almost completely absorb the broth between additions.
Stir in the Parmesan cheese. Continue cooking and adding remaining broth, a cup at a time, for another 8 minutes, or until rice is tender but firm. Remove the pan from the heat.
Stir in the butter and salt as needed. Drizzle with balsamic vinegar and serve immediately.