In the bowl of a stand mixer, combine the warm milk and yeast. Add 1½ cups (180g) of the flour, then incorporate the softened butter, sugar, egg, and salt. Beat using the paddle attachment at medium-low speed for approximately 1 minute, ensuring the mixture is well blended. Scrape down the bowl and paddle before switching to the dough hook attachment. Introduce another 1¾ cups (210g) of flour and mix at medium speed until the dough starts pulling away from the bowl sides, about 2 minutes. If necessary, gradually add the remaining ¼ cup of flour a tablespoon at a time to adjust the dough's consistency; it should be moist but not sticky when touched. Continue kneading with the dough hook for an additional 2 minutes until the dough forms a ball around the hook and pulls from the bowl sides, though some sticking at the bottom is normal. The dough should feel smooth and spring back when pressed. Transfer the dough into a greased bowl, cover with plastic wrap, and let it rise in a warm environment (approximately 75°F) until doubled in size, about 45 minutes to 1 hour. Prepare a 9×13-inch baking dish by greasing with softened butter or applying nonstick spray. Deflate the risen dough gently and place it on a lightly floured surface. Divide into 15 equal portions, each weighing about 55g, keeping the floured side down. Cover the dough pieces with plastic wrap or a kitchen towel to retain moisture while shaping. Shape each piece by pinching the corners to the center, flipping over, and rolling it in a circular motion under your hand to create a smooth, taut ball. Arrange the shaped dough evenly in the prepared pan, cover loosely, and allow to rise again until noticeably puffy, 30 to 45 minutes. After 15 minutes of proofing, preheat the oven to 350°F. Bake the rolls for approximately 20 minutes until deeply golden and puffed. Immediately brush the freshly baked rolls with melted butter. Serve warm or allow to cool before presenting.