Set the eggs and butter out on the counter to take the chill off. If your buttermilk is cold, set the measured buttermilk in a bowl of warm water for about 10 minutes until it reaches room temperature.
Stir the active dry yeast and a pinch of the sugar into the warm water and let it sit for 5 to 10 minutes, until foamy. If it does not foam, your yeast may be old, so start over with fresh yeast.
Add the yeast mixture, buttermilk, remaining sugar, 2 eggs, flour, salt, and softened butter to the bowl of a stand mixer fitted with the dough hook.
Mix on low until no dry flour remains, then knead on medium for 8 to 10 minutes, until the dough is smooth, elastic, and pulling away from the sides of the bowl. By hand, knead for 12 to 15 minutes.
Cover the bowl and let the dough rise until roughly doubled, about 1½ hours at room temperature, or about 45 minutes in an oven set to its 100°F proof setting.
Turn the dough out onto an unfloured counter and divide it into 14 pieces of about 75g each, or see the notes for smaller sizes.
Shape each piece into a tight ball by folding the edges into the center, flipping it seam side down, and rolling it in small circles under a cupped hand to build surface tension.
Line a quarter sheet pan with parchment and arrange 12 rolls in 4 rows of 3, leaving a little space between them so they rise into each other. Bake the last 2 in a small parchment lined pan, or bake all 14 together on a half sheet pan.
Cover loosely and proof for 45 minutes to 1 hour, until puffy and touching, and a gentle poke springs back slowly. If you proof in the oven, take the pan out before preheating.
Preheat the oven to 375°F (190°C).
Whisk the egg with 1 tsp water and gently dab it over the tops of the rolls. Sprinkle with flaky sea salt, if using.
Bake for 20 to 23 minutes, until deep golden brown and an instant read thermometer in the center roll reads about 190°F.
Brush the hot rolls with melted butter right away and let them cool for about 10 minutes before serving.