Using a stand mixer, combine 1 cup of flour, sugar, yeast, and salt. With the machine on low speed, gradually add the six tablespoons of softened butter and mix until the mixture resembles sand, about 2–3 minutes. Slowly pour in the warm water in a steady stream, mixing for about 1 minute.
3 to 3-1/2 cups all-purpose flour, 1/4 cup granulated sugar, 1 (.25) 1/4 oz. envelope active dry yeast, 1/2 tsp. table salt, 10 tbsps. butter, 6 softened, 4 melted, 2/3 cup warm water (about 110 degrees F)
Increase the speed to medium and beat until smooth, about 2 minutes, scraping down the bowl as needed. Finally, add the room temperature egg and beat on low speed for about 1 minute.
1 large egg at room temperature
Using the stand mixer fitted with a dough hook, add 2 cups of flour, 1/4 cup at a time, mixing on low speed. Scrape down the bowl once, and continue mixing until the dough forms a ball, about 2 minutes.
Increase the mixer speed to medium and knead the dough for 5 minutes until it’s smooth, elastic, and slightly sticky. If needed, add up to 1/2 cup more flour (I didn’t need to). Cover the bowl and let it rest in a warm spot for 10 minutes.
On a lightly floured surface, divide the dough into thirds, then cut each third into 12 equal pieces and roll them into balls.
Brush a large, heavy-duty baking sheet with 1 tablespoon of melted butter. Arrange the dough balls about 1 inch apart on the prepared pan. Cover with a kitchen towel and let them rise in a warm spot until doubled in size, about 1 1/2 hours.
Preheat the oven to 400°F.
Bake the rolls for 12–15 minutes, until golden brown. Brush with the remaining 3 tablespoons of melted butter and sprinkle with coarse salt.
coarse salt