Boil 6 cups of water with the salt and the olive oil. Transfer 2 cups to a heatproof glass measuring cup. With the remaining water still boiling, gradually sprinkle in the cornmeal, stirring constantly. Keep adding and stirring, paying particular attention to the corners of the pan so no polenta sticks there and burns. When all the polenta has been added, reduce the heat to a simmer and continue to cook, stirring every 5 minutes, until the polenta is soft, creamy, and tender, about another 30 to 40 minutes, or 10 for the instant. If the polenta becomes too thick (it should be like porridge), add additional hot water.
While the polenta is cooking, use 1 tablespoon of the butter to grease a large baking sheet: chill in the refrigerator.
When the polenta is cooked, stir in the chopped arugula and 6 tablespoons of the butter. Season with salt and white pepper. Pour the polenta onto the chilled baking sheet and smooth it evenly. Smear the remaining 1 tablespoon butter over the surface. Let cool, then cover the sheet with plastic film and refrigerate until firm. Cut into squares or rounds until needed for reheating in olive oil or butter.