In a medium saucepan over medium heat, combine milk, cornmeal, cube of cold butter, sugar and salt. Stir together and cook about 5 minutes. Remove from heat and allow to cool. Mixture will thicken as it cools.

In the bowl of your stand mixer, add active dry yeast, 1/2 cup warm water (110 degrees F), and 1 teaspoon sugar and allow to get foamy, about 10 minutes. Combine cornmeal mixture, yeast, and 2 well-beaten eggs. Add 3 cups flour and mix to make a soft dough. Add one more additional cup of flour and knead on speed 2-4 for 5 minutes. Place dough in bowl that has been greased with olive oil. Cover loosely with plastic wrap and allow to rise for one hour in warm room.

Preheat the oven to 375 degrees F.

Punch down dough and place on a very well floured surface. Work in as much flour as needed to be able to handle dough without it sticking to your hands. Cut into 16 equal sized pieces. Tuck corners under each piece of dough to create a nice dome top and set on a large parchment or silpat lined baking sheet.

Bake rolls in preheated oven for 12 to 15 minutes. Rub tops with butter as soon as you remove them from the oven.

Serve warm as dinner rolls or allow to cool and use as buns (8 large) or slider buns (16).