The tomato puree tints the usually yellow polenta orange, and gives it a rich flavor.
Puree tomatoes with their juice; strain. Place liquid in a large saucepan with the water, bring to a boil, and add salt. Slowly sprinkle in polenta, whisking constantly.
When all the polenta has been added, turn heat down to low and cook, stirring very frequently, for 30 minutes. Stir in pepper and Parmesan.
Pour into an ungreased 9-by-13-inch pan and smooth with a spatula. Cover with plastic wrap and let cool. Refrigerate until cold and firm, at least 2 hours.
Turn polenta out onto a board. Cut into 2-inch squares. Cut each square in half to form 2 triangles. Set aside.
Cut mushroom into 1/2-inch dice. Heat a large saute pan over medium-high heat. Add 1 tablespoon oil and garlic,; cook until lightly browned. Add mushrooms. Keep warm, or reheat when ready to use.
Puree olives with remaining tablespoon of oil and stir into mushrooms. Keep warm, or reheat when ready to use.
Heat a large saute pan and add a scant teaspoon each of oil and butter. Saute polenta triangles for about 2 minutes on each side and briefly on the edges. Add more oil and butter for each batch. Serve hot with a heaping teaspoon of the mushroom mixture on each triangle.
14-ounce can of Italian plum tomatoes can be replaced with 1 1/2 cup of them.