Make the Polenta:

1. Boil the liquids:
   • In a medium saucepan, bring 4 cups of water and 1 cup of milk to a boil. Add a generous pinch of salt.
2. Cook the polenta:
   • Gradually whisk in the polenta, stirring continuously to prevent lumps. Reduce heat to low and cook for 20–25 minutes, stirring often, until the polenta is thick and creamy.
3. Finish with butter and cheese:
   • Remove from heat and stir in the butter and Parmesan cheese. Adjust seasoning with salt and freshly ground black pepper. Cover and set aside.

Poach the Eggs:

4. Prepare the water:
   • Fill a saucepan with 2–3 inches of water. Add 2 tbsp white vinegar and bring to a gentle simmer.
5. Poach the eggs:
   • Crack each egg into a small bowl. Carefully slide each egg into the simmering water. Poach for 3–4 minutes, until the whites are set but the yolks remain runny. Use a slotted spoon to remove the eggs and drain on a paper towel.

Assemble the Dish:

6. Serve:
   • Spoon the creamy polenta onto plates or bowls. Place a poached egg on top of each serving.
7. Garnish and enjoy:
   • Season with additional salt, freshly ground black pepper, and a sprinkle of Parmesan cheese if desired. Serve immediately.