In a medium saucepan, bring 4 cups of water or broth to a gentle simmer over medium heat. Gradually whisk in 1 cup of coarse or medium-ground polenta, ensuring there are no lumps. Reduce the heat to low, and let it simmer for about 20-25 minutes, stirring frequently until creamy and thickened. If desired, add a splash of milk or cream, along with butter and Parmesan cheese, until melted and blended into a comforting mixture.

While the polenta simmers, fill a separate pot with water and add a splash of white vinegar. Heat the water over medium until it reaches a gentle simmer, avoiding a rolling boil. Crack each egg into small bowls, then create a light whirlpool in the simmering water using a spoon. Gently slide the eggs into the center, cooking them for about 3-4 minutes, or until the whites are set, and the yolk remains runny.

Once the polenta reaches a creamy consistency, spoon generous amounts into warm serving bowls. Carefully place a poached egg on top of each bowl of creamy polenta, allowing the yolk to nestle in. If using, sprinkle freshly chopped parsley or chives over the eggs for a pop of color and freshness. Serve immediately.