In a 2 1/2-quart or larger saucepan, bring the milk, water, 2 teaspoons of salt, and pepper to a low boil, being careful not to allow the liquid to boil over. Working quickly, vigorously whisk in the cornmeal (to avoid lumps). Reduce the heat to low and cook for 20 minutes, stirring frequently with a heat-resistant rubber spatula. Cook's Note: Make sure to stir along the sides and bottom of the pan so the polenta does not stick. Once the polenta is tender, remove the pan from the heat and stir in the 3 tablespoons butter and the mascarpone cheese.
Cut two pieces of parchment paper to line a 13 by 9-inch rimmed baking sheet. Lay one piece of parchment on the sheet and brush 1 tablespoon of the room-temperature butter over it. Pour the polenta onto the buttered parchment and spread the polenta into a smooth, even layer. Brush the other piece of parchment with the remaining 1 tablespoon butter and place the parchment, butter side down, on top of the polenta. Refrigerate the polenta until it is firm, at least 2 hours or up to 2 days in advance.
Preheat a grill to medium-high.
Mix the Parmesan cheese, lemon juice, remaining 1/2 teaspoon salt, and extra-virgin olive oil in a small bowl and reserve.
Trim the polenta, cutting off about 1 inch on each side to form straight edges, and cut it into 6 squares. Halve each square to form 2 triangles. You should have 12 triangular pieces.
Make sure the grill is well cleaned, and oil it with some of the olive oil. Brush the top of the polenta squares generously with olive oil, and place the polenta, oiled side down, on the grill.
Brush the tops with olive oil. After about 3 minutes, rotate the polenta 90 degrees to make grill crosshatch marks. Cook's Note: If you try to turn or move the polenta before a crust has formed on the grill side, it will stick. Cook for an additional 3 minutes. Turn the polenta squares over and repeat, cooking until they are nicely browned and crisp, about 6 minutes longer.
Move the grilled polenta to a serving platter and immediately top with the reserved Parmesan-lemon mixture. Serve hot.