Heat oven to 375°F. Grease 2 cookie sheets with shortening, or spray with cooking spray.
If using crescent rolls: Unroll dough; separate into 2 rectangles, about 7x6 inches each. Press perforations to seal. If using dough sheet: Unroll dough; cut into 2 rectangles, about 7x6-inches each.
Sprinkle cherries evenly over dough, pressing into dough. Brush rectangles with half of the butter. In small bowl, mix sugar and cinnamon. Sprinkle half of sugar mixture evenly over dough.
Starting with one short side of each rectangle, roll up; pinch edge to seal. With serrated knife, cut each roll into 3 slices. Place each slice, cut side down, between 2 sheets of plastic wrap or waxed paper; roll or press each into 5-inch round. Remove plastic wrap; place 2 inches apart on cookie sheets.
Brush rounds with remaining butter; sprinkle with remaining sugar mixture. Cover rounds with plastic wrap or waxed paper; press or roll sugar mixture into dough.
Bake 14 to 16 minutes or until golden brown. Immediately remove from cookie sheets. Serve warm or cool.