Combine the yeast, sugar, and 2 tablespoons of the oil in the bowl of an electric mixer fitted with a dough hook. Add the water. With the mixer on low speed, beat the mixture for about 4 minutes to dissolve the yeast. If the yeast mixture doesn’t begin to foam after a few minutes, it means it’s not active and will have to be replaced.
In a separate large mixing bowl, combine the flour, 1/2 cup plus 2 tablespoons of the cornmeal, and the salt. Add this mixture to the yeast mixture. Mix on low speed until it lightly comes together, then increase the speed to medium and beat until the mixture pulls away from the sides of the bowl, forms a ball, and climbs slightly up the dough hook.
Remove the dough from the bowl. Coat the bowl with the remaining teaspoon vegetable oil. Return the dough to the bowl and turn it to oil all sides. Cover the bowl with plastic wrap, set in a warm, draft-free place, and let rise until doubled in size, about 2 hours.
Meanwhile, brown the sausage in a medium-size skillet over medium heat for about 4 minutes. Drain well on paper towels. Set aside to cool to room temperature.
Remove the dough from the bowl and turn it onto a lightly floured surface. Using your hands, gently roll and form it into a narrow loaf about 24 inches long. Cut the dough into 18 equal portions (each about 2 1/2 ounces). With the palm of your hand, roll the portions on a slightly floured surface to form small round rolls.
Line a baking sheet pan with parchment or waxed paper and sprinkle it with the remaining 2 tablespoons cornmeal.
Place the rolls about 1 inch apart on the paper. Using a pointed knife, make a slit in the top of each roll. With your thumb and forefinger, spread the dough open to make a small cavity about 1 inch deep and 2 inches wide. Spoon 1 tablespoon of the cheese into each cavity, then top with 1 tablespoon of the sausage, pressing the mixture gently into the cavity. Cover the rolls with plastic wrap and let rise in a warm, draft-free place until doubled in size, about 30 minutes.
Heat 4 inches of oil, or enough to submerge the breads, in a deep pot or electric fryer to 360ºF. deep-fry the stuffed breads, two to three at a time, in the hot oil for about 3 minutes, turning them with a metal spoon or spatula to brown them evenly. Drain on paper towels. Serve warm.