Grease loaf pan with butter.

In a large saucepan, bring water, 1 tablespoon oil and 1/2 teaspoon salt to a boil. Slowly add polenta in a thin stream, whisking; reduce heat to medium and cook, whisking constantly, for 2 minutes. Reduce heat to low and cook, stirring often with a wooden spoon, until thickened and tender, about 40 minutes. Season with 1/2 teaspoon pepper, then remove from heat.

Transfer polenta to prepared pan and spread evenly with a rubber spatula. Let polenta cool completely in pan on a wire rack. When polenta is cool, turn out onto a cutting board. Sprinkle cheese on top, pressing lightly with fingers to help cheese adhere. Cut polenta into 1-inch cubes.

In a large non-stick skillet, heat remaining tablespoon oil over medium heat. Add polenta cubes, cheese-side down, and cook until browned, 4 to 6 minutes. Flip and cook until bottoms are browned, about 2 minutes more. Divide cubes among 4 bowls.

In a medium saucepan, bring broth just to a boil. Divide broth among bowls. Serve warm.