Apple Pie à la Mode Ice Cream Sandwiches

Preheat oven to 375°F. Line 4 large baking sheets with parchment paper. Whisk together flour, baking soda, baking powder, cream of tartar, and salt in a medium bowl. Combine shortening, butter, and 1 cup granulated sugar in bowl of a stand mixer fitted with the paddle attachment. Beat on high speed until light and fluffy, about 2 minutes. Reduce speed to low; beat in egg, egg yolk, and vanilla bean paste until incorporated. Gradually add flour mixture, beating just until dough comes together.

Whisk together cinnamon and remaining 2 tablespoons granulated sugar in a shallow bowl. Shape dough into 42 (1-inch) balls. Dredge balls in cinnamon mixture, and arrange 3 inches apart on prepared baking sheets. Bake cookies in preheated oven in the upper and lower thirds of oven, until light golden around edges, 7 to 9 minutes. Transfer cookies on baking sheets to wire racks to cool completely, about 15 minutes.

Melt butter with olive oil in a large skillet over medium. Add bay leaf, and cook 15 seconds. Stir in apples, brown sugar, lemon juice, cinnamon, allspice, nutmeg, cloves, and salt. Cook, stirring occasionally, until apples are very soft and begin to caramelize, about 25 minutes. Scrape apples onto a baking sheet or large plate to cool completely; discard bay leaf.

Working quickly, fold cooled apple mixture into softened ice cream in a large bowl. (If ice cream becomes too soft, place in freezer until firm enough to scoop, about 30 minutes.) For each ice cream sandwich, scoop about 11/2 tablespoons ice cream mixture onto flat side of half of cookies; cover with remaining cookies. Wrap in plastic wrap, and freeze until ice cream mixture is just firm, about 30 minutes. Serve immediately. Assembled ice cream sandwiches can be wrapped in plastic wrap and frozen up to 3 days.