Make the chile oil: Combine the chopped chiles, olive oil and 1/4 teaspoon salt in a small saucepan over low heat and cook 5 minutes. Remove from the heat and let cool completely. Transfer to a bowl, cover and refrigerate up to 1 week. (You will have about 1 1/4 cups chile oil.)
Cook the polenta: Bring 3 cups cold water, the milk, 1 teaspoon salt and 1/2 teaspoon pepper to a boil in a medium saucepan over high heat. Gradually whisk in the polenta; whisk constantly until the mixture is smooth and begins to thicken, about 5 minutes. Reduce the heat to medium low and simmer, stirring often, until tender and thickened but still creamy, about 30 minutes. Add more hot water as needed if the mixture seems too dry. Stir in the butter and cheese.
Meanwhile, make the braised greens: Heat the olive oil in a large high-sided saute pan over medium heat. Add the onion and cook until softened, 4 minutes. Add the garlic and cook, stirring occasionally, until slightly golden brown, about 2 minutes. Add the greens, season with salt and pepper and toss. Reduce the heat to medium, stir in 1 cup water, cover and cook until wilted and soft, about 20 minutes. Remove the lid during the last 5 minutes to cook off any additional liquid. Stir in the red wine vinegar just before serving.
Poach the eggs: Heat 3 cups water with the white vinegar until simmering in a large high-sided frying pan. Break each egg into a cup, then gently add to the water. Poach until the yolks are nearly set, 4 to 5 minutes. Remove the eggs from the pan with a slotted spoon to drain the liquid and place on a plate. Season with salt and pepper.
To serve, spoon some of the polenta into bowls, top with some of the greens and a poached egg per bowl. Drizzle with the chile oil and garnish with the chopped parsley.