In a large bowl, whisk ¾ cup all-purpose flour with the sugar, caraway seeds, salt, and yeast.

In a small saucepan, heat the milk and shortening to 120° to 130°.

Add the warm milk mixture and the egg to the flour mixture and blend; beat for 3 minutes on medium speed.

Add the rye flour and beat well; then add enough of the remaining all-purpose flour to make a stiff batter.

Grease a muffin pan; spoon the batter into the prepared muffin cups to ⅔ full and let the batter rise, uncovered, for about 25 to 30 minutes in a warm place.

In the meantime, preheat the oven to 400°.

Bake the rolls for 12 to 15 minutes or until done.

Brush tops with melted butter.