In a small saucepan, bring the milk, water, cream and butter to a boil. Reduce the heat to medium and gradually whisk in the cornmeal. Cook, whisking constantly, until thickened, about 8 minutes. Stir in the Grana Padano.

Remove the stems. Using a teaspoon, gently scoop out some of the flesh to create a shallow cavity in each mushroom.

Brush the mushrooms with olive oil and arrange them on a parchment-lined baking sheet. Spoon the polenta into the mushroom caps, leaving a little room at the top.

Bake until the mushrooms are tender and the polenta is hot, about 10 minutes. Serve warm, topped with additional Grana Padano.