Preheat oven to 350°F. Line a 12-cup muffin pan with paper or foil liners. In a medium bowl, whisk the flour, cornstarch, baking powder, and salt. Set aside. In a large bowl, using a handheld electric mixer, beat the sugar, eggs, and vanilla extract at medium-high speed, until smooth and thickened slightly (about three minutes). Add the butter and oil, and beat until incorporated. Add the dry ingredients and milk in an alternating pattern, beating well between additions. Carefully pour the batter into the lined muffin cups, filling them about 2/3 full. Bake the cupcakes in the center of the oven for 20-23 minutes or until a toothpick inserted in the center comes clean. Let the cupcakes cool slightly, then transfer them to a wire rack to cool completely. For icing, beat butter until smooth at medium speed with a handheld electronic mixer. Add powdered sugar, vanilla, and salt. Beat the mixture at low speed until combined. Add milk and beat until smooth (about two minutes). Finally, beat in as much strawberry jam to meet your taste bud's demands. To finish, top cupcakes with the icing, as well as some strawberry slices.