In a small saucepan over the lowest heat, melt the butter. Once melted, stir in the milk, and check the temperature to be sure it reaches at least 95 degrees F.
While the butter is melting, whisk the flour, sugar, yeast, and salt in a large bowl. Set aside.
Once the butter mixture reaches 95 degrees F, transfer it into the bowl of a stand-up mixer, with the dough hook attached. Add the egg, then the dry ingredients to the bowl. Start on the "stir" speed to allow everything to combine, then slowly turn up to the "4" speed and mix for 5 minutes.
Turn off the mixer, and scrape down the sides. The dough will be sticking to the sides at this point, and that's okay. Stir on "4" for another 5 minutes.
If the dough is still very sticky and not pulling away from the sides of the bowl, add additional flour one tablespoon at a time until the dough pulls away from the sides and comes together.
Lightly grease a large bowl (you can use the one that the dry ingredients were mixed in), and set aside.
Turn the dough out onto a lightly floured surface, and gently form it into a ball. Place the ball into the greased bowl, and cover with a kitchen towel or plastic wrap. Set the bowl in a warm place for 45 minutes, or until doubled in size.
Roll the dough out again onto a lightly floured surface, and divide into eight equal portions if using a 10.5 x 7.5 casserole dish like mine. If using a 9x9 baking dish, divide the dough into nine equal portions. Roll the portions to form dough balls, and pinch the bottoms to seal them shut.
Grease your preferred baking dish, and sprinkle with at least one tablespoon of the everything bagel seasoning. (Add more if you'd like more seasoning on the bottom of your rolls). Place the rolls in your preferred baking dish, seam side down. Cover again with a kitchen towel or plastic wrap, and allow to rise an additional 30 minutes.
Turn the oven to 350 F.
Melt the butter in a small saucepan, and brush the butter over the tops of the rolls. Sprinkle the rolls with remaining everything bagel seasoning (or more if you like), and bake in the oven for 25-30 minutes at 350 F, until golden and lightly browned.
Carefully remove from the oven, allow to cool for a few minutes, then serve warm!