Heat a medium heavy nonstick skillet over medium-high heat. Add the sausage and cook, stirring, for 5 minutes. Add the onions and cook, stirring, until lightly golden, about 5 minutes. Add the jalapeños and garlic and cook, stirring often, for 2 minutes. Using a slotted spoon, transfer the mixture to paper towels to drain and cool.
Combine the yeast, sugar, water, and 2 tablespoons of the oil in a small bowl. Let stand for about 5 minutes, then whisk to dissolve the yeast and sugar.
Combine 5 cups of the flour, 3/4 cup of the cornmeal, the salt, the Cheddar, the sausage-onion mixture, and the yeast mixture in the bowl of a large electric mixer fitted with a dough hook. Mix on low speed until the dough comes together and forms a ball, about 3 minutes. If the dough is too moist, add 1/4 cup flour and mix to incorporate, about 1 minute. Check the dough consistency again, and add some or all of the remaining 1/4 cup flour if needed. After the dough forms a ball, knead with the dough hook for 5 minutes longer.
Coat a large mixing bowl with the remaining 1 teaspoon vegetable oil. Remove the dough from the mixer, place it in the bowl, and turn it to oil all sides. Cover with plastic wrap or a clean damp towel. Let stand in a warm, draft-free place until it doubles in size, about 2 hours.
Sprinkle the remaining 1 tablespoon cornmeal over two large baking sheets.
Remove the dough from the bowl and divide it into 3 equal portions. Roll each portion out under your hands on a lightly floured surface into a log about 24 inches long. Cut each log into 8 equal portions. One at a time, gently push down the dough while rolling it under your hand in a circular motion to form a smooth ball. Place the balls about 11/2 inches apart on the prepared baking sheets.
Snip the top center of each roll with kitchen shears to form an X. Cover the rolls with plastic wrap. Let stand in a warm, draft-free place, until they double in size, about 30 minutes.
Meanwhile preheat the oven to 400°F.
Remove the plastic wrap. Bake the rolls, switching the positions of the sheets from top to bottom halfway through the baking time, until golden brown, about 20 minutes. Serve hot.
Yield: 24 rolls