Whisk the melted butter, half-and-half, yeast, and 1 teaspoon of the sugar in a large liquid measuring cup until the yeast dissolves. Whisk in the egg and egg yolk.
In the bowl of a stand mixer fitted with the dough hook, mix the flour, remaining sugar, and salt on low speed until combined, about 15 seconds. With mixer running on low, add the half-and-half mixture in a steady stream and mix until the dough comes together. Increase the speed to medium and mix until the dough is smooth and comes away from the sides of bowl, about 6 minutes. Turn the dough out onto a clean surface and knead to form a smooth, round ball. Transfer the dough to a greased bowl, cover with plastic wrap, and place in a warm, draft-free spot. Let rise until doubled in size, about 1 hour.
Turn the dough out onto a lightly floured surface and roll into a 12-inch circle. Spread the softened butter over the dough, sprinkle with the salt, and cut into 12 wedges using a pizza cutter. Roll each wedge starting at the wide end, and arrange (with pointed tip of dough underneath each roll) on a baking sheet lined with parchment paper. Cover loosely with plastic wrap and return to a warm, draft-free spot until doubled in size, about 1 hour. Make-Ahead Note: At this point, the rolls can be refrigerated for up to 24 hours before baking.
Preheat oven to 350 degrees F. Whisk the egg white and water in a small bowl. Remove the plastic wrap and brush each roll twice with the egg white mixture. Bake until golden brown, about 20 minutes, rotating the baking sheet halfway through cooking. Cool rolls on the baking sheet for 10 minutes before serving. Leftover rolls can be stored in an airtight container at room temperature for up to 2 days.