Bring Liquid to a Simmer: In a medium-sized heavy-bottomed saucepan, bring the water or broth (and milk/cream, if using) to a simmer over medium heat. Add a pinch of salt to the liquid.

Whisk in Polenta Gradually: Slowly pour the polenta into the simmering liquid in a thin, steady stream while whisking constantly. This is important to prevent lumps from forming.

Reduce Heat and Simmer: Once all the polenta is added, reduce the heat to low, so the mixture is gently simmering. Continue to whisk constantly for the first few minutes to ensure no lumps form.

Cook and Stir Regularly: Switch to a wooden spoon or spatula and continue to cook the polenta, stirring frequently (every few minutes), for about 25-30 minutes, or according to the package instructions for your specific polenta. The polenta should thicken and become creamy and smooth. If it becomes too thick, add a little more hot water or broth to reach your desired consistency.

Stir in Butter and Cheese: Once the polenta is cooked and creamy, remove it from the heat. Stir in the butter and Parmesan cheese (if using) until they are melted and fully incorporated. Season with salt and pepper to taste. Keep the polenta warm while you prepare the poached eggs. You can cover the pot and place it over very low heat, stirring occasionally, or keep it in a warm oven (around 200°F/95°C).

Prepare the Poaching Water: Fill a wide, deep skillet or large saucepan with about 3-4 inches of water. Add the white vinegar and a pinch of salt. Bring the water to a gentle simmer – you should see small bubbles rising to the surface, but not a rolling boil.

Crack Eggs into Individual Bowls: Crack each egg into a small, separate bowl or ramekin. This makes it easier to gently slip them into the water.

Create a Swirl (Optional but Recommended): Using a spoon, gently swirl the simmering water to create a gentle vortex in the center. This helps the egg white wrap around the yolk and form a neater poached egg.

Poach the Eggs: Carefully and gently pour each egg from its bowl into the center of the swirling water (or directly into the simmering water if not swirling). Poach 2-3 eggs at a time, depending on the size of your pan, to avoid overcrowding and lowering the water temperature too much.

Cook to Desired Doneness: Poach the eggs for 3-4 minutes for runny yolks, or 4-5 minutes for firmer yolks. The egg whites should be set, but the yolks should still be soft and liquid. You can gently poke an egg white with a spoon to check for firmness.

Remove and Drain: Once the eggs are poached to your liking, use a slotted spoon to carefully remove them from the water. Gently blot the bottom of each egg on a paper towel to remove excess water.

Spoon Polenta into Bowls: Divide the creamy polenta among bowls or plates.

Top with Poached Eggs: Place a poached egg on top of each serving of polenta.

Garnish and Serve Immediately: Garnish with freshly grated Parmesan cheese (if desired), freshly ground black pepper, and chopped fresh parsley or chives. Serve immediately while the polenta and eggs are warm and delicious.