Grease and flour a cake mold pan (such as a bunny cake or train cake mold) using vegetable shortening.
Sift flour, baking powder, and salt in a bowl.
Mix 1 1/3 cup milk, lemon extract, 1 1/4 teaspoons vanilla extract, and orange extract in a separate bowl.
Beat unsalted butter in a large mixing bowl with an electric mixer on medium speed until butter is creamy and smooth, 1 to 2 minutes. Beat in white sugar and continue beating until the mixture is light and fluffy, about 5 more minutes. Scrape the sides of the bowl occasionally.
Beat in eggs, one at a time, beating each egg in thoroughly before adding the next.
Reduce mixer speed to low and beat in flour mixture in several additions, alternating with milk mixture; begin and end with flour mixture. Do not overbeat. Scrape bowl occasionally.
Pour batter evenly into prepared cake pan.
Bake in the preheated oven until lightly browned and a toothpick inserted into the middle of the cake comes out clean, 30 to 45 minutes.
Place cake in pan on a wire rack and allow to cool for 15 minutes before inverting pan to remove cake. Allow cake to completely cool on wire rack, 3 to 4 hours.
Beat vegetable shortening, 1/2 cup butter, confectioners' sugar, 2 tablespoons milk, and 1 teaspoon clear vanilla extract in a bowl until frosting is smooth and creamy.
Pack the frosting into a heavy resealable plastic bag; force the frosting into one corner of the bag. Clip a small corner off the plastic bag, seal the bag, and pipe the frosting onto the cooled cake in desired pattern.