Pour warm milk (110-115 degrees F) into the bowl of stand mixer. Add sugar and yeast. Stir, and let it stand for about 10 minutes until frothy.
Set up the dough hook attachment in the stand mixer. With the mixer on low speed, mix the eggs, 1/4 cup melted and cooled unsalted butter, honey, and salt. Increase to medium speed, and add flour 3 cups flour, 1 cup at a time. Add the remaining flour in 1/4 cup increments as needed, until a dough has formed. If it’s sticking to the bowl too much, continue adding flour in ¼ cup increments. I ended up using 3 1/2 cups. The dough should be slightly sticky, moist, and manageable.
Transfer to a floured surface and knead by hand about 4 minutes, sprinkling more flour as necessary. The dough should be moist, slightly sticky, and should spring back slowly when poked. Do not over-knead or the bread will be tough.
Form into a ball and place on a lightly oiled bowl (olive oil is fine), making sure to coat the dough with some of the oil. Cover with plastic wrap and let it rise in a warm environment for about 1 hour, until doubled in size.
Punch down the dough and transfer to a clean surface. Cut the dough in half, and then each half into 6 equal pieces.
Shape into balls, pinching the bottom. Arrange in a greased 9x13 baking pan. Cover with plastic wrap and let it rise for another hour in a warm environment, until doubled in size.
Preheat oven to 350 degrees F while the dough is rising.
Brush bread with melted butter. Bake for about 15-20 minutes. Let cool slightly.
Combine butter with honey with a whisk. Brush some over the top and/or use it as a spread.