Bring 4 cups water to a boil in a medium-sized heavy pot — enameled cast iron is ideal. Add 1½ teaspoons salt, then slowly add the polenta in a stream while whisking. When all the polenta has been added, stir and turn down the heat to a very low simmer. Cook 45 minutes, stirring every 10 minutes. If you find the polenta getting too thick, add a bit of water as needed and lower heat.
While the polenta cooks, bring a pot of salted water to a boil. Trim off any thick stems from your greens and parsley drop them into the water. Cook for 2 minutes. Drain, reserving about a cup of the cooking water.
Put the greens in a blender or food processor with 1 tablespoon each of olive oil and butter, adding a small splashes of the cooking water if needed, to get it going. Puree until fairly smooth. Stir in the lemon juice.
Stir the remaining olive oil, butter and half of the cheese into the polenta. Put 1 or 2 tablespoons cheese in the bottom of each serving bowl before spooning in the polenta. Top each serving with some of the pureed greens for each person to stir into their bowl. Sprinkle with additional cheese at the table.