In the bowl of a stand mixer, combine 3 cups of flour, yeast, sugar, and salt. Stir to mix. Add the warm milk, softened butter, and egg.
Attach the dough hook and mix on the lowest speed until the flour is fully incorporated, scraping down the sides as needed.
Increase the speed to medium and mix for 2 minutes.
Gradually add ½ cup of flour at a time, blending with the dough hook after each addition. Continue mixing at medium speed for 1–2 minutes until a soft dough forms. If needed, add up to an additional ½ cup of flour until the dough is slightly sticky and pulling away from the sides of the bowl.
Transfer the dough to a lightly greased bowl, cover with a towel, and let it rise at room temperature for 30 minutes.
Remove the towel and gently punch down the dough to release air.
Divide the dough into 20 equal portions and roll them into balls.
Place the dough balls into a lightly greased 9x13-inch baking dish. Cover with a towel and allow them to rise for another 30 minutes.
Preheat the oven to 375°F. Bake for 15–17 minutes or until golden brown and fully cooked. If the tops brown too quickly, cover them loosely with foil.
Remove from the oven and brush the warm rolls with melted butter and honey. Serve immediately.