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.
Photograph by Con Poulos
Recipe courtesy Food Network Magazine