To a stand mixer bowl with a dough hook attached, add 3 cups of flour, yeast, and salt.
In a saucepan add milk and ½ cup butter. Heat* until butter is melted. Add sugar and stir to dissolve.
Add milk mixture to flour and mix on low speed. Scrape sides as needed. Beat egg and add to dough mixture. Mix at medium speed for 2 minutes.
At low speed, gradually add the remaining 1 cup flour. Turn speed to medium and mix for 2 minutes until a dough is formed. If dough is sticking to sides, add flour 1T at a time to create a ball, but only enough to do so.
Transfer dough to a greased bowl, cover and allow to rise for 60 min. It should be doubled in size.
Punch down, pulling the sides of dough to the center and press down. Set on a lightly floured surface, cover with bowl, and allow to rise 15 min.
Cut into two pieces. Roll each piece to a size of about 9 x 11. Melt 1 T butter, and brush rolled out dough.
Cut each section in half lengthwise creating 4 - 4½ x 11 inch pieces. Fold each in half lengthwise and leave a bit of the buttered side exposed. Cut each piece into 4 even sized pieces. This will create 16 pieces.
Grease a 9x13 inch pan with 1 Tablespoon butter. Turn rolls over and place four in a row (the longer side will be parallel with the long side of the dish)
Cover with a towel and rise for about 45 min.
Preheat oven to 350. Bake 20-25 min until golden brown. Remove and cool slightly, then brush each top with butter.
Notes
*Alternately heat in the microwave just until the butter melts in the milk.