Meanwhile, we use a flat whisk to combine the egg, sugar (reserving a teaspoon or so), 1 tablespoon of the oil, and the salt in a large bowl.
When the water is approximately 110° F, we sprinkle on the yeast and the reserved teaspoon of sugar. We set this aside for 5 minutes until it is foamy.
While the yeast is “proofing”, we measure out the flour onto a large piece of parchment paper.
We stir the yeast into the large bowl with the whisk until it is combined. Gradually, we add the flour, pouring right from the parchment paper into the bowl. We keep stirring the dough with the whisk. When we have about ½ cup of flour yet to add, it is too difficult to stir with the whisk. With very clean hands, we remove the clinging dough from the whisk and add the remaining flour to the dough with our hands. The dough is soft and a little sticky, so we do the best we can to make a smooth ball of dough in the bowl.
With a pastry brush, we spread the remaining ½ tablespoon of oil on the top of the dough. We cover the bowl with plastic wrap and set it aside in a warm spot in our kitchen for an hour. We want the dough to double in bulk.
When the hour is about up, we melt the butter in a microwaveable cup in the microwave for 20 seconds. We just want it to be melted. Lightly, we brush the butter on the bottom and sides of a 9-inch round cake pan with the cleaned pastry brush. The remaining butter is reserved for the top of the rol ls.
When the dough is ready, we punch it down and then, with floured hands, we divide the dough into 12 pieces. This is the hardest part of the recipe, really, to try and get the rolls to be about the same size.
Each piece of dough is rolled gently in our floured hands and then arranged in the pan. We put 9 of the rolls (they are about an inch and a half in diameter) around the circumference of the pan and three fit in the middle. We cover the pan loosely with the plastic wrap and set it aside for a second rising in the same location in our kitchen. This takes another half an hour. We want the rolls to swell up in the pan.
While the rolls rise, we preheat the oven to 400° F.
When the rolls have risen, we discard the plastic wrap and reheat the butter quickly in the microwave. Then, we brush the remaining butter on the tops of the rolls.
We put the pan in the center of the oven and cook for 20 minutes. We want the rolls to be nicely browned and cooked through.
We set the pan on a trivet for 5 minutes and then unmold to a plate. Everyone pulls a roll off and enjoys it hot!