Heat the oven to 375°F with a rack in the lower-middle position.
In a large oven-safe saucepan or small (4- to 5-quart) Dutch oven, combine the cornmeal, 1 teaspoon salt and 5½ cups water, then whisk to combine.
Bring to a gentle simmer over medium, stirring often, then place uncovered in the oven and cook for 45 minutes.
Remove from the oven, whisk the polenta, then return, still uncovered, to the oven.
Cook until the polenta is thick and creamy, another 15 to 30 minutes.
Remove from the oven, whisk until smooth, cover and let stand for 5 minutes.
Taste and season with salt and black pepper, then cover and set aside while you cook the shrimp.