Bring water and milk to a boil in a medium saucepan over high heat. Reduce the heat to low and whisk in polenta, nutritional yeast, garlic, and salt. Cook, constantly whisking, until the mixture thickens and begins pulling away from the pan's sides, about 5 minutes.
Spoon the polenta into ramekins coated with canola cooking spray. Cool for five minutes and freeze the ramekins for 45 minutes until set.
While the polenta is in the freezer, heat butter in a large skillet over medium-high heat until sizzling. Add the mushrooms. Cook, undisturbed, until bottoms brown, about 4 minutes. Continue to cook, occasionally stirring, until tender and browned—about 8 minutes. Reduce the heat to medium.
Add the red onion, thyme, and garlic. Cook, stirring, for 2 minutes. Add the tomatoes and cook, stirring, for an additional minute. Stir in the vegetable broth, vinegar, butter, and sprouts. Cook for 1 minute, stirring. Remove from heat and set aside. Do not cover.
Heat the olive oil in a large nonstick skillet over medium-high, swirling to coat. Add the polenta cakes and cook until golden brown and crisp, about 4 minutes per side. Spin the cakes by quarter turns every minute to help with even browning. In the last 2 minutes of cooking the cakes, bring Ragu up to a simmer, stirring. Place the cooked cakes on plates; top with the ragu. Garnish with shaved Parmesan and serve.