Add buttermilk, ½ stick of butter, and sugar to a small saucepan and heat over medium low, stirring occasionally with a spatula. Heat just until butter has melted, then remove from heat. Use a thermometer to check the temperature. It should be warm but not hot and read somewhere between 100 and 110°F. If your mixture registers hotter than that, let it cool on the counter for a bit, or pop it in the fridge for a couple of minutes, then recheck with your thermometer. Otherwise, if the liquid is too hot it could kill your yeast.
You can also warm your buttermilk, butter, and sugar in the microwave. Just use a microwave-safe bowl and cook 1 minute at a time, stir, check, repeat until your butter has fully melted. Don't forget a microwave shield or your liquid will spatter.
Add the flour, yeast, and salt to the bowl of a stand mixer fitted with the dough hook. Add the buttermilk mixture, followed by 2 eggs. Mix on medium speed for about 5-7 minutes, or until dough is smooth, elastic, and comes free from the sides of the bowl. (If you don’t have a stand mixer and dough hook, you can mix this in a bowl with a spatula or wooden spoon, and then knead with your hands for about 10 minutes.)
Preheat oven as low as it will go (mine is 170°F) just for one minute, then turn it off (the oven will be just warm enough for proofing your dough). Brush a large bowl with softened butter. Transfer the dough to the bowl and cover with a kitchen towel. Place in the oven to proof for 1 hour (the dough will double in size).
Divide the dough in half, cutting it with a bench scraper or sharp knife. Cut each half in half and those halves in half. You’ll want to have 15 equal pieces of dough (for 15 rolls). I recommend weighing each with a kitchen scale if you want, to ensure they’re all the exact same size, about 70g each. If you have any dough leftover, cut off little pieces and add to each roll.
To shape your rolls, you want to allow some friction between the dough and your work surface. Lightly flour your work surface. Cup your hand around the dough ball, resting the side of your hand on your work surface as you gently roll in quick, tight, circular motions until you have a smooth, round shape (don't worry about the bottom as much). As you're rolling, cover your other dough portions with a kitchen towel so they don't dry out.
Brush a 9×13-inch baking dish with softened butter. Arrange each of your rolls in the pan in three rows of five across. Cover with a dish towel and let sit someplace warm for the rolls to proof (I like stashing mine in the microwave), for 1 hour (if the rolls have not doubled in size by then, you may need to allow another 30 minutes of proofing time).
Once the rolls have finished proofing, preheat your oven to 350°F. Break your remaining egg into a small bowl and whisk well. Brush each roll with your egg wash (you will not use all of it) and place rolls in the oven to bake for about 20-23 minutes, or until golden brown. Once the rolls are done, brush them with more softened butter and top with a sprinkling of flaky sea salt.