In small saucepan over medium heat, heat milk, butter and honey just until butter is melted. Remove from heat and let sit for 5 minutes (temperature should be 120-140 degrees F).
Meanwhile, in stand mixer fitted with dough hook, combine 1 1/2 cups four, salt and yeast. Add milk mixture and mix until incorporated. Add eggs and mix on medium speed until smooth, about 3 minutes.
Add remaining flour 1/4 cup at a time until dough clears the bowl but is still slightly sticky to the touch. You may not need all of the flour, or you may find you need a bit more. Knead with dough hook until smooth and elastic, about 5-7 minutes.
Shape dough into ball and place in buttered bowl. Cover with plastic wrap and put in warm place to rise until doubled in size, about one hour.
Punch down dough and divide into 12 equal pieces. Shape into rounds and place in buttered 9x13-inch baking dish. Cover with plastic wrap and let rise again until an indentation remains when lightly pressed with your finger, about 30-45 minutes.
Preheat oven to 375°F. Brush tops of rolls with egg wash and bake until golden brown on top, about 20-25 minutes. Remove rolls from oven and immediately brush with butter. Serve warm.