In a large saucepan, bring 4 cups of water or broth to a gentle simmer over medium heat. Gradually whisk in 1 cup of coarse polenta, stirring continuously to avoid lumps. Lower the heat to medium-low and cook, stirring often for about 15-20 minutes, or until the polenta is thickened and creamy. Stir in 2 tablespoons of butter and 1/2 cup of Parmesan cheese until fully melted and combined.

In a separate pot, fill it with water about 3 inches deep and bring it to a gentle simmer, adding a splash of white vinegar. Crack each of the 4 eggs into small bowls, then carefully slide them into the simmering water one at a time. Cook for 3-5 minutes, depending on your preference for firmness.

Once your creamy polenta is ready, spoon it generously into individual serving bowls. Nestle a poached egg atop each mound of polenta, allowing the yolk to contrast beautifully with the creamy base. Finish with a sprinkle of freshly chopped parsley or chives for a burst of color. Serve immediately.