In the bowl of a stand mixer fitted with the paddle attachment, combine flour, sugar, yeast, and salt. With mixer on low speed, add 3 eggs, egg yolks, and warm buttermilk, beating until mixture comes together, 2 to 3 minutes. (If mixture remains too dry and crumbly, add more buttermilk, 1 tablespoon at a time.) Increase mixer speed to medium-high, and beat for 6 minutes.
Switch to the dough hook attachment. With mixer on medium speed, add butter in three additions, letting each addition incorporate before adding the next. Increase mixer speed to medium-high, and beat until a smooth and elastic dough forms and pulls away from sides of bowl. (If dough does not pull away from bowl, add more flour, 1 tablespoon at a time.)
In a small bowl, combine cheese, chives, and green onion. In another small bowl, whisk together 1 tablespoon water and remaining 1 egg.
Turn out dough onto a heavily floured surface, and divide into 3 equal portions. Working with one portion at a time, roll dough into a 16x12-inch rectangle. Brush one long side of dough with egg wash. Sprinkle one-third of cheese mixture over dough. Starting at opposite long side, roll up dough, jelly-roll style, pressing seam to seal. Place on a sheet pan. Repeat twice with remaining dough and remaining cheese mixture. Braid all three ropes together, and tuck ends under. Place braid in prepared pan. Cover and let rise in a warm, draft-free place (75°) until doubled in size, 1½ to 2 hours. (Alternatively, dough can be made 1 day in advance, and refrigerated overnight.)
Preheat oven to 350°.
Brush top of dough with remaining egg wash. Bake for 30 minutes. Cover loosely with foil, and bake until an instant-read thermometer inserted in center registers 190°, about 25 minutes more. Let cool in pan for 15 minutes; remove and let cool on a wire rack. Serve slightly warm.
Recipe by Taste of the South at https://tasteofthesouthmagazine.com/buttermilk-cheddar-chive-bread/