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For Polenta:
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Grease an 8 x 8 inch baking dish (with sides) with butter.
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Add water and oil to a large saucepan and heat over high heat until you have a boil. Reduce heat to low and pour in cornmeal while whisking constantly to prevent lumps from forming. Bring the mixture back to a boil and continue cooking for about 5 minutes until the polenta is tender and thick. Be sure to stir your polenta constantly during the five minutes of cooking.
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Remove the pan from the heat and whisk in butter, cheese and parsley. Add salt and pepper to taste and spoon the cooked polenta into your prepared pan, spreading the polenta to even it out in the pan. Chill in the refrigerator until form (at least 20 minutes or overnight).
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To make the ragout:
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Heat a large skillet over medium-high heat and add butter and oil. When the butter is melted and the butter foam subsides, add onion and garlic. Sitr with a wooden spoon until the onion is translucent and the garlic is fragrant. This will take about 6 minutes. Add mushrooms, sage, and thyme leaves. Continue cooking for about 10 minutes until the mushrooms have released their liquid and the liquid has evaporated. Add wine and cook until the wine has evaporated. Add tomatoes and cook, stirring until the ragout thickens. This will take about 10 minutes. Taste and adjust seasoning with salt and pepper and sugar. Stir in heavy cream.
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To assemble final dish:
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Melt unsalted butter in a large saucepan and put rectangular pieces of chilled polenta into the hot pan. Cook on one side for about 5 minutes until polenta begins to brown. Turn polenta over and cook on the other side for about 5 minutes.
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Put polenta slices on oven-proof serving plates, spoon on a generous dollop of the hot mushroom ragout and top with shredded mozzarella and Asiago cheeses. Heat for a few minutes under your broiler (or microwave) and serve garnished with extra cheese and chopped parsley.