In a medium bowl, whisk together the water (between 100°F and 110°F), one tablespoon of granulated sugar, and active dry yeast. Set aside to proof for 5-10 minutes.
As the yeast is proofing, whisk together the flour, salt, and remaining sugar in the bowl of a large mixer fitted with the dough hook or paddle attachment. After the yeast has activated, add it and the softened butter to the flour.
Beat/mix on low speed for 30 seconds, then scrape down the sides of the bowl with a silicone spatula. Continue kneading the dough for a full 5-7 minutes at medium-low (2x) speed. The dough should form a smooth, elastic ball that pulls away from the sides without sticking too much. If the dough is too sticky, add a little more flour, one tablespoon at a time.
First Rise: Lightly spray a large, clean glass bowl with cooking spray or lightly coat it with oil. Place the dough in the bowl, gently turning to coat all sides with oil. Cover the bowl with plastic wrap or a clean kitchen towel. Allow the dough to rise in a warm (no warmer than 90°F), draft-free area for about 1-2 hours or until it has doubled in size.
Grease a 9x9-inch baking pan (or a 9x13-inch baking dish if making 12 rolls). Once the dough is ready, punch it down to release the CO2 and turn it out on a clean work surface (do not grease or flour the countertop). Divide the dough into 9-12 equal pieces (a bench scraper is helpful for cutting dough). (I used a scale to get nine equally sized pieces that weighed approximately 3.35 oz each.) Gently flatten each piece with your hand into a rough disk shape. To shape into balls, gently cup the piece of dough in your palm. Working in a circular motion, use the edges of your palm and fingers to tuck the dough under itself, creating a smooth, tight ball. Pinch the seams at the bottom of the balls and tuck them underneath. Arrange in the prepared baking pan.
Second Rise: Cover the baking dish with plastic wrap or a clean kitchen towel and let them rise for 30-60 minutes, or until puffy.
Adjust the oven rack to a lower position and preheat the oven to 350°F.
Whisk the egg and water in a small bowl until no lumps remain. Once the rolls are ready, brush the tops with the egg wash just before baking.
Bake the rolls for 20-25 minutes or until the tops spring back within a second of being gently pushed with your finger (always check the middle roll first since it will take the longest to bake). Rotate the pan halfway through cooking. If the tops start to brown too quickly, tent the pan loosely with aluminum foil. Cool for a few minutes before serving.
Serve with flaky sea salt or brush the tops with melted butter, if desired.