In a large, ovenproof skillet, heat 1 tablespoon olive oil over medium heat. Add the onion, season with salt and cook until softened, about 5 minutes. Transfer to a large bowl; reserve the skillet.
In a small saucepan, bring 2 1/3 cups water and 1/4 teaspoon salt to a boil. Whisk in the polenta, lower the heat to medium-low and whisk until thickened, about 5 minutes. Stir in the cream, remove from the heat, then stir in the cheese; season with salt and pepper and keep warm.
Preheat the broiler. Add the pork and parsley to the onion; shape into four 4-inch-wide patties. In the reserved skillet, heat 1 tablespoon olive oil over medium heat. Add the patties and cook, turning once, until browned, 6 to 7 minutes. Spoon the polenta on top and broil until golden, about 5 minutes.
In a saucepan, bring 1/3 cup water and the remaining 1 tablespoon olive oil to a boil over medium heat. Add the beans, cover and cook for 3 minutes; drain. Season with salt and pepper. Serve with the patties.
