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Take half of the butter and slice it as thin as you can, add the slices to the bottom of a stand mixer. Add the sugar and salt to the mixer.
Warm the milk in a microwave safe bowl in the microwave in 15 second intervals. It should be 105-110°F (do not warm to over 110°F). Pour 1 cup of the warm milk into the bowl with the butter, no need to stir, just let it sit.
Add the yeast to the remaining 1/4 cup milk and give it a stir. Let it sit for about 5 minutes, until foamy. Add this mixture to the mixer, along with the beaten egg.
Using the hook attachment, start stirring in the flour 1 cup at a time, scraping down the sides as needed. Do this until you've added 3 & 1/2 cups of flour. Once the mixture resembles a dough, turn the speed to medium-high and start to knead it. If it doesn’t start to clean the sides of the bowl, add a little more flour at a time (up to a 1/2 cup) until the sides of the bowl are clean - if the dough is still attached at the very bottom of the bowl, that’s okay. Knead for 3 minutes.
In a large bowl, add the vegetable oil and swirl it around. Take the dough out of the mixer and add it to the bowl with the oil, turn it around in the oil so it is coated. Cover with plastic wrap and let it rise in a warm place for 1-2 hours until doubled in size.
Melt the remaining 1/4 cup of butter. Brush a 9x13 baking dish with some of the butter (you will only need a small amount). Set aside.
Punch down the dough to release the air bubbles. Divide the dough into 15 sections. (I like to weigh the dough then divide by 15 to see how many ounces each roll should be and make sure they're uniform in size.)
Take each dough ball with your hands and stretch it out, tuck the ends of the dough around itself to form a smooth ball and pinch the seams together, gently roll it between your hands to form a ball.
Place the balls seam side down into the prepared baking dish, 5 rows of 3 rolls. Cover again with plastic wrap and let them double in size, about 1 hour.
Preheat the oven to 350°F. Take the plastic wrap off the baking dish and bake for 20-25 minutes until the tops are golden brown and when tapped the rolls sound hollow. Immediately brush the rest of the butter all over the hot rolls.Let cool and serve.