To prepare cookies: Whisk all-purpose flour, pastry flour, baking soda, cinnamon, ginger, cloves and salt in a medium bowl. Beat butter, oil, molasses, sugar and egg in a large bowl with an electric mixer on low speed until combined. Add half the flour mixture and beat on medium speed until incorporated. Add the remaining flour mixture and beat until just combined.Advertisement
Divide the dough in half. Place one half on a 12-inch-long sheet of parchment paper dusted with flour; shape into a disk. Top with another dusting of flour and a second sheet of parchment. Roll the dough out between the parchment into a circle about 1/8-inch thick. Place the dough in the paper on a baking sheet. Freeze on the baking sheet until cold and firm, at least 30 minutes and up to 1 day. Repeat with the remaining dough.
To shape & bake cookies: Position a rack in middle of oven; preheat to 350 degrees F. Line 2 large baking sheets with parchment paper.
Working with one portion of dough at a time, remove from the freezer and place on a clean work surface. Remove the top sheet of parchment and, using a a mini gingerbread house cookie cutter, cut out enough cookies to make 12 houses. Transfer to the prepared baking sheets with a spatula, spacing about 1 1/2 inches apart. (If the dough gets too soft, or starts to stick to the cutter, freeze until firm again. As you cut out cookies, set aside the scraps. Shape all the scraps back into a disk and reroll between parchment. Freeze and save for additional houses or another use.)
Bake the cookies on the center rack, one pan at a time, until browned on the bottom, 10 to 12 minutes. Let stand for 5 minutes, then transfer to a cutting board. Even out the pieces by trimming the outer edges, including the "door" that rests the house on the lip of the cup, if desired. Transfer to a wire rack to cool.
To prepare icing & decorate cookies: Whip egg white and cream of tartar in a medium bowl with an electric mixer on high speed until foamy. Add confectioners' sugar in batches, mixing well after each addition. Beat on high until the mixture is stiff and glossy, 1 to 2 minutes. Scoop the icing into a pastry bag with a small tip or into a small plastic bag with a tiny tip of one corner snipped off and follow the assembly instructions that come with your cutter. Houses are ready to use when the icing has hardened, about 4 hours.
To make ahead: Store, covered, at room temperature for up to 3 days.