Brush an 8 1/2-by-4 1/2-inch loaf pan lightly with oil. Pour Soft Polenta into pan; smooth top. Cover with plastic wrap, and refrigerate until chilled and firm to the touch, about 3 hours and up to 3 days.
Meanwhile, make mushroom ragout: In a large skillet, heat oil over medium. Add onion; cook, stirring often, until soft, about 5 minutes. Add mushrooms; season generously with salt and pepper. Cover and cook until mushrooms release their juices, about 10 minutes.
Uncover pan; raise heat to high. Cook, stirring often, until liquid has evaporated and mushrooms are brown, 2 to 3 minutes. Add rosemary, tomato paste, and 2 cups water. Simmer until sauce has thickened, 10 to 15 minutes. Stir in vinegar.
Meanwhile, heat broiler. Invert polenta onto a cutting board: cut into 12 to 14 slices, each 1/2 inch thick. Brush a rimmed baking sheet with oil. Arrange slices in a single layer; brush with oil. Broil 4 inches from heat, without turning, until golden and browning in spots, 10 to 20 minutes. Divide among plates; top with mushroom ragout, and sprinkle with goat cheese. Serve with Wilted Spinach and Cherry Tomatoes, if desired.
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