In a medium bowl, mix the yeast, sugar and water; let stand for 5 minutes. Mix in the buttermilk and eggs. In a large bowl, whisk 5 cups of the flour with 2 teaspoons of salt. Make a well in the center of the flour; add the buttermilk mixture. Stir with a wooden spoon until nearly blended; beat in the 1 1/2 sticks of softened butter until a soft dough forms.
Scrape the dough onto a work surface sprinkled with the remaining 3/4 cup of flour. Knead until smooth, 3 minutes. Transfer the dough to a lightly oiled bowl, cover and let rise in a warm, draft-free spot until doubled in bulk, about 1 hour.
Meanwhile, melt 3 tablespoons of the butter in a large saucepan. Add the onions and cook over high heat, stirring, until softened, 7 minutes. Reduce the heat to moderate and cook, stirring occasionally, until golden, 20 minutes; add a bit of water if the onions stick. Stir in the nutmeg and cloves. Season with salt. Transfer to a bowl to cool.
Butter two 9-inch round cake pans. Melt the remaining 3 tablespoons of butter. Turn the dough out onto a lightly floured surface, punch it down and divide in half. Roll 1 piece of the dough out to an 8-by-16-inch rectangle; a long side of the rectangle should be facing you. Keep the other piece covered. Spread half of the onions all over the dough to within 1-inch of the top. Starting at the bottom edge, roll as tightly as possible into a cylinder. Pinch the edge to seal, then brush with half of the melted butter. Cut crosswise into 12 slices, wiping the knife after each slice. Arrange the rolls, cut side up, in 1 of the pans, 9 around the outside and 3 in the center. Wipe the work surface and repeat with the remaining dough, onions and melted butter. Cover the rolls with plastic; let rise until almost touching the rims of the pan, about 1 hour.
Preheat the oven to 425°. Brush the rolls with the egg white. Bake for 35 minutes, or until risen and golden on top. Let cool in the pans for 5 minutes, then turn out onto a wire rack. Serve hot or at room temperature.
Make Ahead: The baked rolls can be wrapped in foil and stored at room temperature for up to 2 days. Reheat in a 350° oven, then unwrap and bake until slightly crusty. Alternatively, freeze for up to 1 month. Thaw before reheating.