Remove the mixing bowl from the stand mixer, add flour, yeast, and salt. Mix to combine. Create a well in the center of the flour mixture (a small hollow spot in the middle for the wet ingredients to go).
In a medium bowl, add warm milk, butter, and honey and whisk until butter is melted (milk should be hot enough to melt butter, but not too scalding - should be lukewarm by the time the butter is melted). Add to the well in the dry ingredients. Do not mix yet.
In another bowl, add egg and egg yolks and beat for 1-2 minutes until mixed well. Add this to liquid ingredients in the bowl.
Place the mixing bowl onto the stand mixer, add the dough hook attachment, and mix on medium speed for 5-6 minutes - or until elastic and smooth.
While dough is mixing, add a teaspoon of vegetable oil to a large metal bowl and rub around the inside of the bowl.
Transfer the dough to the bowl, cover with a tea towel and let rise for 45 minutes.
While the dough is rising, butter a 9x13 baking dish and set aside.
Punch down and divide the dough into 3 equal circles. Divide each circle into 5 parts to make 15 equal pieces.
Roll each piece into a ball making sure to stretch the dough and pinch it underneath to ensure a smooth top.
Place each ball into the prepared baking dish. Cover with a tea towel and let rise for 1-1.5 hours, or until doubled in size.
Preheat oven to 350°F. Brush the tops of the rolls with an egg wash and bake for about 20-25 minutes, or until golden brown.