Grease an 8-inch square baking dish. In a heavy, medium saucepan, bring 3 cups water and 1 teaspoon salt to a boil over high heat. Add the polenta in a thin stream, whisking constantly. Lower the heat and simmer, stirring constantly, until thick, 5 minutes; season with salt and pepper. Spread evenly in the prepared baking dish.
In a large skillet, heat 2 tablespoons butter over medium-high heat. Add the mushrooms and cook, stirring frequently, until golden, about 5 minutes. Add 2 tablespoons water, then add the kale in batches, allowing each batch to wilt slightly before adding the next. Cook, stirring, until tender, about 8 minutes; season with salt and pepper.
In a small saucepan, melt the remaining 1 1/2 tablespoons butter over medium heat. Whisk in the flour and cook, stirring constantly, until golden, about 3 minutes. Slowly pour in the milk, whisking, and cook, stirring constantly, until the sauce thickens, about 10 minutes; season with salt and pepper.
Preheat the oven to 400 degrees . Top the polenta with the kale mixture. Spoon the cream sauce evenly over the kale mixture and top with the provolone. Bake until warmed through, about 15 minutes. Increase the temperature to 450 degrees and bake until golden, about 5 minutes.