Combine the yeast, sugar, melted butter, egg and milk in the bowl of an electric mixer fitted with a dough hook. Beat on low speed for 1 minute. Add the salt, rye flour, and all-purpose flour. Beat at low speed until all of the flour is incorporated, about 1 minute. Then beat at medium speed until the mixture forms a ball, leaves the sides of the bowl, and climbs up the dough hook. Remove the dough from the bowl. Using your hands, form the dough into a smooth ball. Lightly oil a bowl. Place the dough in the bowl and turn it to oil all sides. Cover with plastic wrap and set aside in a warm, draft-free place until it doubles in size, about 1 hour.
Preheat the oven to 350 degrees F.
Remove the dough from the bowl and invert it onto a lightly floured surface. Pat the dough into a rectangle about 3/4-inch thick. Roll up the dough, beginning with the short side and stopping after each full turn to press the edge of the roll firmly into the flat sheet of dough to seal. Press with your fingertips. Tuck and roll so that any seams disappear into the dough. Cut the dough into 1-inch pieces. Roll each piece of dough into a smooth round ball. Line a baking sheet with parchment or waxed paper. Place the rolls on the baking sheet, 1/2-inch apart. With a pastry brush, brush the beaten egg evenly over the bread. Cover with plastic wrap and set aside in a warm, draft-free place until it doubles in size, about 1 hour. Bake until lightly brown, 30 to 35 minutes. Remove from the oven and cool on a rack. Serve warm with butter.