Combine initial wet and dry ingredients: In a mixing bowl fitted with a dough hook attachment, combine the warm milk mixture, the activated yeast mixture, granulated sugar, salt, and 1 cup of the all-purpose flour. Beat on high speed until the ingredients are well combined and the batter is smooth.
Add egg: Add the egg to the mixing bowl and beat until it is fully incorporated into the batter.
Gradually add remaining flour: Slowly add the remaining 2 ½ cups of all-purpose flour, a little at a time, while the mixer is on low speed. Continue mixing until the dough comes cleanly away from the sides of the mixing bowl and the dough hook attachment. This may take a few minutes.
First rise: Lightly flour the dough and turn it out into a large mixing bowl that has been well greased. Turn the dough to coat it lightly with oil. Cover the bowl with plastic wrap or a clean kitchen towel and let the dough rest in a warm place to rise for 1 hour, or until it has doubled in size.
Shape the rolls: Place the risen dough onto a lightly floured surface and roll it out until it is approximately ½ inch thick. Use a knife or pizza cutter to cut the dough into 12 even rectangular shapes.
Form the roll shape: Take each rectangle of dough and fold the short ends underneath to create a rounded roll shape. Place the shaped rolls onto a baking sheet lined with parchment paper or a silicone mat, leaving some space between each roll.
Second rise: Cover the baking sheet with a clean kitchen towel and let the rolls sit in a warm place to rise slightly for another 20 minutes.
Preheat oven: About 10 minutes before the end of the second rise, preheat your oven to 400°F (200°C).
Bake the rolls: Bake the rolls in the preheated oven for 12-15 minutes, or until they are lightly browned on top.
Brush with butter (optional): If desired, brush the tops of the warm baked rolls with melted butter immediately after removing them from the oven.
Make the honey cinnamon butter: While the rolls are baking, combine the softened unsalted butter, cinnamon, honey, and confectioners sugar in a small bowl. Mix with a fork or spoon until all the ingredients are completely combined and the butter is smooth and creamy.
Serve: Serve the warm Texas Roadhouse-style rolls immediately with the honey cinnamon butter. Enjoy!