Bring 7 cups water to a simmer in a large saucepan. Combine 4 cups of the hot water and the milk in a heatproof bowl (the bowl should fit over the saucepan). Gradually whisk in the polenta and 2 teaspoons salt. Cover the bowl tightly with foil and set over the simmering water; do not let the bowl touch the water. Cook until very thick, 20 minutes. Meanwhile, brush a 9-inch square baking dish with olive oil.
Stir the cheese and butter into the polenta. Spread in the prepared dish and refrigerate until set, at least 2 hours or overnight.
Preheat the oven to 450 degrees F. Slice the chilled polenta into eighteen 4 1/2-by-1-inch sticks and transfer to a baking sheet lined with parchment paper. Brush the sticks with olive oil and sprinkle with salt. Bake until golden, 30 to 35 minutes, turning halfway through.