Bring 4 cups salted water or broth to a boil in a saucepan. Whisk in 1 cup coarse polenta slowly to avoid lumps. Reduce heat to medium-low and simmer 25-30 minutes, stirring every 2 minutes, until polenta pulls away from the sides and looks thick and creamy.
1 cup coarse polenta, 4 cups water or broth, 1 teaspoon salt
Remove polenta from heat and stir in ½ cup grated Parmesan, 1 tablespoon butter, and ½ teaspoon black pepper until cheese melts. Pour into a shallow dish or plate, smooth the top, and refrigerate 2-3 hours until firm enough to slice cleanly.
0.5 cup Parmesan cheese, 3 tablespoons butter or olive oil, 0.5 teaspoon black pepper
Cut chilled polenta into 8 squares or wedges about ½ inch thick. Pat dry with paper towels to remove surface moisture—this helps edges crisp during frying.
1 cup coarse polenta
Heat 2 tablespoons butter or olive oil in a nonstick or cast iron skillet over medium heat. Place 3-4 polenta slices in the pan with space between them. Cook 3-4 minutes per side until golden and crispy, then transfer to a plate. Repeat with remaining slices.
3 tablespoons butter or olive oil
In the same skillet, add more oil if needed and crack 4 eggs. Cook on medium-low heat 2-3 minutes for runny yolks, or longer for set yolks. Place one egg on each polenta cake and serve immediately with extra Parmesan and black pepper.
4 large eggs, 0.5 cup Parmesan cheese, 0.5 teaspoon black pepper
