Preheat the oven to 350°. Line a 12-cup muffin tin with paper or foil liners.
In a medium bowl, whisk the flour with the cornstarch, baking powder and salt.
In a large bowl, using a handheld electric mixer, beat the sugar with the eggs and vanilla extract at medium-high speed until smooth and thickened slightly, about 3 minutes. Add the butter and oil and beat until incorporated, scraping the bottom and side of the bowl. Add the dry ingredients and milk in 3 alternating batches, beating well between additions. Carefully pour the batter into the lined muffin tins, filling them about two-thirds full.
Bake the cupcakes in the center of the oven for 20 to 23 minutes, until springy and a toothpick or cake tester inserted in the center comes out clean. Let the cupcakes cool slightly in the muffin tin, then transfer them to a wire rack to cool completely.
In a medium bowl, using a handheld electric mixer, beat the softened butter at medium speed until smooth. Add the confectioners' sugar, vanilla extract and salt and beat the mixture at low speed just until combined. Increase the mixer speed to medium and beat until smooth. Add the milk or heavy cream and beat until light and fluffy, about 2 minutes.
Mix the frosting with strawberry jam to taste. Spread the frosting on the cooled cupcakes and serve.
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