Start by preheating your oven to 400 degrees. Preheat oven to 400 degrees. Lightly spray a 9 x 9 inch baking dish with nonstick spray.
In the bowl of a stand mixer fitted with a dough hook, combine warm water, vegetable oil, yeast, and sugar. Stir gently to combine the ingredients and dissolve the yeast. Cover with a towel and let the mixture rest for about 15 minutes.
Once the yeast mixture is ready, add salt, a lightly beaten egg, and the flour to the bowl. Set the stand mixer to medium speed and mix the ingredients together until the dough starts to come together. Continue to knead with the dough hook for 2-3 minutes, or until the dough is smooth, elastic, and no longer sticky. You may need to scrape down the sides of the bowl as needed to ensure everything is fully incorporated.
Once the dough has been kneaded, divide it into 9 equal portions. Roll each portion into a smooth ball and place them in a lightly greased 9x9-inch baking dish. Arrange the dough balls so they are just touching each other, as this will help them bake together in the perfect shape.
Cover the pan with a clean kitchen towel or plastic wrap and let the dough rest for about 10 minutes. This will allow the rolls to rise slightly before baking.
After the dough has rested, place the baking dish in the preheated oven and bake for 10 to 15 minutes, or until the rolls are golden brown on top. The rolls should have a soft, fluffy texture on the inside with a lightly crisped exterior.
Once the rolls are done baking, remove them from the oven and let them cool for a few minutes before serving before brushing with melted butter. Serve warm.