Heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add the onions, 1 teaspoon of the sugar, and ½ teaspoon of salt. Cook, stirring occasionally, until golden brown, 15 to 18 minutes.
3 tbsps. olive oil, 3 medium yellow onions, 1/4 cup plus 1 teaspoon granulated sugar, 1 tsp. kosher salt
In a large bowl, combine 2 cups of warm tap water (105°F to 115°F). Sprinkle the yeast over the water and let stand until foamy, 4 to 5 minutes. Whisk in the eggs, 4 tablespoons of the butter, the remaining ¼ cup of sugar, and remaining ½ teaspoon of salt. Add the flour and three-quarters of the onions, mixing until a sticky dough forms. Brush the top of the dough with the remaining 1 tablespoon of oil. Cover the bowl with plastic wrap and let rise in a warm place until doubled in size, about 1 hour.
2 cups warm water, 2 (0.25 oz.) packages active dry yeast, 2 large eggs, 6 tbsps. butter, 6 cups all-purpose flour
Butter a 13×9-inch baking pan. Punch down the dough. With well-floured hands, form the dough into 16 balls* and place them in the prepared pan, spacing evenly. Sprinkle with the remaining onions. Cover the pan loosely with plastic wrap and let rise in a warm place until doubled in size, 30 to 40 minutes.
Preheat the oven to 400°F.
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. If the tops and onions are browning too quickly, loosely cover with foil halfway through baking to prevent burning. Let cool in the pan for 5 minutes before serving.