Directions

1. Heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add the onions, 1 teaspoon of the sugar, and 1/2 teaspoon salt and cook, stirring occasionally, until golden brown, 15 to 18 minutes.
   
2. Place 2 cups warm water in a large bowl; sprinkle with the yeast and let stand until foamy, 4 to 5 minutes. Whisk in the eggs, 4 tablespoons of the butter, the remaining 1/4 cup sugar, and 1 teaspoon salt. Add the flour and three-quarters of the onions and mix until a sticky dough forms. Brush the top of the dough with the remaining tablespoon of oil. Cover the bowl with plastic wrap and let rise in a warm place until doubled in bulk, about 1 hour.
   
3. Butter a 9-by-13-inch baking pan. Punch the dough down. With well-floured hands, form the dough into 16 balls and place in the prepared pan, spacing evenly. Sprinkle with the remaining quarter of the onions. Cover the pan loosely with plastic wrap and let rise in a warm place until doubled in bulk 30 to 40 minutes.
   
4. Heat oven to 400 degrees. Remove the plastic wrap and brush the dough with the remaining 2 tablespoons of butter. Bake until the rolls are golden, 18 to 22 minutes (tent with foil if the tops brown too quickly). Cool in the pan for 5 minutes before serving.