1. To prepare cookies, weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, 1 cup sugar, baking soda, cardamom, and salt in a food processor; process until blended. Add almond paste and butter; process until blended. Combine juice and oil. With processor on, slowly pour juice mixture through food chute; process until dough forms a ball.
2. Divide dough in half. Working with 1 portion at a time, gently press dough into a 4-inch square on heavy-duty plastic wrap. Cover with additional plastic wrap. Roll each half of dough, still covered, to a 1/4-inch thickness. Chill 1 hour.
3. Preheat oven to 375°.
4. Working with 1 portion of dough at a time, remove top sheet of plastic wrap; turn dough over. Remove remaining plastic wrap. Cut with a 2 1/2-inch star-shaped cookie cutter into 24 stars; place stars 2 inches apart on 2 baking sheets lined with parchment paper. Bake at 375° for 8 minutes or until lightly browned. Cool on pans 5 minutes. Remove from pans; cool on wire racks. Repeat procedure with remaining half of dough.
5. To prepare icing, combine 2 cups sugar and milk, stirring with a whisk until smooth. Spread about 1 teaspoon icing over each warm cookie; chill 1 hour.