In a large saucepan, heat the buttermilk and cheese over low heat, stirring until cheese is melted (mixture will appear curdled). Cool to 110° to 115°. In a large bowl, dissolve sugar and yeast in warm water. Add the buttermilk mixture, eggs, salt and 3 cups flour. Beat until smooth. Stir in enough remaining flour to form a soft dough.
Turn onto a floured surface; knead until smooth and elastic, about 6-8 minutes. Place in a greased bowl, turning once to grease top. Cover and let rise in a warm place until doubled, about 1 hour. Meanwhile, for topping, combine cheddar cheese and chilies; set aside.
Punch dough down; turn onto a lightly floured surface and divide in half. Roll each portion into an 18x12-in. rectangle. Spray one half with cooking spray. Top with cheese mixture and remaining dough.
Cut into twelve 1-1/2-in. strips. Cut each strip in half and twist. Place 1 in. apart on greased foil-lined baking sheets. Sprinkle with Parmesan cheese.
Bake at 375° for 15-20 minutes or until lightly browned. Remove to wire racks. Serve warm. Refrigerate any leftovers.