In a medium bowl set over a saucepan of simmering water, whisk together polenta and 2 cups water. Cook until water is absorbed and polenta is cooked, about 5 minutes. Set aside, stirring occasionally, until cool.
In a separate medium bowl, stir together the flour, sugar, 3 tablespoons salt, baking soda, egg yolk, milk, and remaining 1/3 cup water. The mixture will be lumpy. Stir in the polenta until smooth.
In a large Dutch oven, heat the oil until it registers 350 degrees F on a deep-fry thermometer. Pierce the sausage with chopsticks. Do not pierce all the way through. Working in batches, dip the sausage into the polenta batter until evenly coated. Place the battered sausage in hot oil (still on sticks); cook, turning occasionally, until the corn dogs float to the top and are evenly browned, 4 to 5 minutes.
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.