3 tablespoons butter, divided
Bring water and salt to a boil in a large saucepan; pour polenta slowly into boiling water, whisking constantly until all polenta is stirred in and there are no lumps. Reduce heat to low and simmer, whisking often, until polenta starts to thicken, about 5 minutes. Polenta mixture should still be slightly loose. Cover and cook for 30 minutes, whisking every 5 to 6 minutes. When polenta is too thick to whisk, stir with a wooden spoon. Polenta is done when texture is creamy and the individual grains are tender.
Turn off heat and gently stir 2 tablespoons butter into polenta until butter partially melts; mix 1/2 cup Parmigiano-Reggiano cheese into polenta until cheese has melted. Cover and let stand 5 minutes to thicken; stir and taste for salt before transferring to a serving bowl. Top polenta with remaining 1 tablespoon butter and about 1 tablespoon freshly grated Parmigiano-Reggiano cheese for garnish.