To make the polenta, generously butter a 13-by-9-inch baking dish.

In a saucepan over high heat, combine the milk, broth and bay leaf and bring to a simmer. Remove from the heat, cover and steep for 20 minutes. Discard the bay leaf.

Return the pan to high heat and bring the liquid to a boil. Gradually add the polenta, whisking constantly until smooth. Reduce the heat to medium-low and cook, stirring frequently, until the polenta is very thick, about 10 minutes. Stir in the cheese and butter. Adjust the seasoning with salt and pepper.

Transfer the polenta to the prepared baking dish and, using your hands, press the polenta evenly to the edges. Refrigerate until firm, at least 3 hours and preferably overnight.

When ready to serve, cut the polenta into 20 squares.

To make the mushroom ragout, in a large, deep fry pan over medium-high heat, melt 2 tablespoons of the butter with 2 tablespoons of the olive oil. Add the mushrooms and cook, stirring often, until tender and browned, about 10 minutes.

Add the shallots and vinegar and cook, stirring often, until the shallots are tender, about 2 minutes. Season lightly with salt and pepper. The mushrooms can be cooked 1 day ahead and will taste even better the next day. Let cool, then cover and refrigerate. Rewarm the mushrooms over medium heat and proceed as directed below.

If serving immediately, add the broth and simmer for 1 minute. Stir in the cream and about half of the parsley. Adjust the seasoning with salt and pepper. Remove from the heat, cover and keep warm.

When ready to serve, preheat the oven to 300°F.  

In a large fry pan over medium-high heat, melt the remaining 1 tablespoon butter with the remaining 2 tablespoons olive oil. Working in batches, add the polenta squares and cook until browned, about 2 minutes per side. Transfer to a baking sheet and keep warm in the oven while you brown the remaining squares.

Arrange the polenta squares on a platter. Top with the mushroom ragout, and sprinkle with the cheese and the remaining parsley. Eat up!