How to Make It
Bring water and milk to just below a boil in a medium saucepan over high. Reduce heat to low, and whisk in polenta, cheese, 1/2 teaspoon of the garlic, and 1/8 teaspoon of the salt. Cook, whisking constantly, until mixture thickens and begins to pull away from sides of pan, about 3 minutes.
Spoon polenta mixture into an 8-inch square baking pan lined with plastic wrap. Top with another layer of plastic wrap, and press mixture into an even layer in pan. Freeze until set, about 45 minutes.
While polenta sets, heat 1 tablespoon of the butter in a large skillet over medium-high until sizzling. Add mushrooms; cook, undisturbed, until bottoms have browned, about 4 minutes. Continue to cook, stirring occasionally, until tender and browned all over, about 8 minutes.
Reduce heat to medium. Add shallot, thyme, and remaining 1/2 teaspoon garlic. Cook, stirring often, until fragrant, 1 to 2 minutes. Stir in vegetable broth, balsamic, and remaining 2 tablespoons butter. Continue to cook, stirring constantly, until sauce is well blended, about 1 minute. Remove from heat; stir in Brussels sprout leaves and cherries, and cover to keep warm.
Turn polenta mixture out onto a cutting board, and cut into 4 (3-inch) circles using a round cutter. (Reserve remaining polenta mixture for another use.) Heat oil in a large nonstick skillet over medium-high; swirl to coat. Add polenta cakes to hot oil, and cook until golden brown and crisp, about 4 minutes per side.
Place 1 polenta cake on each of 4 plates; top each with 3/4 cup of the cherry-mushroom ragu.