1. Butter a 9-inch cake pan using 1 tablespoon softened butter. Butter a large bowl; set aside. Stir together yeast, sugar, and water in a small bowl; let mixture stand until foamy, about 5 minutes. Stir until dissolved. Stir in buttermilk and egg.

2. Mix 2 3/4 cups flour and 1 1/2 teaspoons salt in the bowl of an electric mixer fitted with the dough hook. Make a well in center. Pour in buttermilk mixture; mix to combine. Add 6 tablespoons softened butter; mix on medium-high speed until a soft dough forms, about 10 minutes.

3. Scrape dough onto a lightly floured work surface; sprinkle with remaining 2 tablespoons flour. Knead dough until smooth, about 5 minutes. Transfer to buttered bowl. Cover dough with a clean kitchen towel; let rise in a warm place until doubled in bulk, about 1 hour.

4. Melt remaining 4 tablespoons softened butter in a large saucepan over medium-high heat. Add onions; raise heat to high, and cook, stirring often, until soft, about 5 minutes. Reduce heat to medium; cook, stirring, until golden brown, about 30 minutes. Stir in nutmeg. Season with 1/2 teaspoon salt. Let cool.

5. Punch down dough, and turn out onto a lightly floured work surface. With a lightly floured rolling pin, roll dough into a 17-by-10-inch rectangle, and brush with 3 tablespoons melted butter. Spread onions evenly over dough. Starting on 1 long side, roll dough into a log. Press seam to seal. Cut into about 12 slices, about 1 1/4 inches thick each. Arrange slices, cut sides up, in buttered pan, and brush with remaining 2 tablespoons melted butter. Cover loosely with plastic wrap. Let rise in a warm place until doubled in bulk, about 50 minutes.

6. Preheat oven to 375 degrees. Bake rolls until golden brown, about 35 minutes. Immediately invert and unmold rolls onto a wire rack. Serve warm.