Using a standing mixer, combine 1 cup flour, the sugar, yeast and salt. With the machine on, gradually add the softened butter at low speed until the mixture is the consistency of sand, 1 to 2 minutes. Mix in the warm water in a slow, steady stream for about 1 minute. Increase the speed to medium and beat until smooth, about 2 minutes; scrape down the bowl. Beat in the egg at low speed for about 1 minute.
Using the mixer fitted with a dough hook, mix in 2 cups flour, 1/4 cup at a time, on low speed, scraping down the bowl once, until the dough forms a ball, about 2 minutes. Increase the speed to medium and beat for 5 minutes until smooth, elastic and slightly sticky, adding up to 1/2 cup more flour if needed. Cover the bowl and let stand in a warm spot for 10 minutes.
Brush a heavy-duty baking sheet with 1 tablespoon melted butter. On a lightly floured surface, divide the dough into thirds, then cut each third into 12 equal pieces and roll into balls. Arrange the dough balls about 1 inch apart on the prepared pan. Cover with a kitchen towel and let rise in a warm spot until doubled in size, about 1 1/2 hours.
Meanwhile, preheat the oven to 400 degrees . Bake the rolls until golden-brown, 12 to 15 minutes. Brush with the remaining 3 tablespoons melted butter and sprinkle with the coarse salt.