Place mushrooms in a large baking dish, gill-side up. Drizzle olive oil and balsamic vinegar over mushrooms. Crush thyme between your fingers; sprinkle over mushrooms.
Bake in the preheated oven until tender, about 20 minutes. Transfer to a plate using tongs. Leave any accumulated juices in the baking dish.
Bring water and half-and-half to a boil in a large pot. Add polenta in a steady stream, whisking constantly. Reduce heat and cook, stirring constantly, until thickened, about 10 minutes. Remove from heat. Add Swiss cheese, sea salt, and pepper; stir until cheese is melted.
Pour hot polenta into the same baking dish used to bake the mushrooms. Smooth the top with a spatula. Arrange mushrooms, gill-side up, in the polenta. Sprinkle Parmesan cheese on top.
Bake in the preheated oven until mushrooms are soft, about 20 minutes. Remove from oven and garnish with truffle salt.