Preheat oven to 180°C (350°F). Grease and flour a standard muffin tin.
In a medium bowl, sift together flour, whole-wheat flour, cornstarch, baking powder and salt.
In a stand mixer fitted with the paddle attachment, cream the butter and sugar until light and fluffy. Add the eggs one by one, and the vanilla extract. Slowly add one third of the flour mixture and half the milk. Mix until incorporated. Add the second third of the flour mixture and the other half of the milk. Finish with the final third of the flour mixture. Mix until all the ingredients are well combined. The batter is smooth and fluffy.
Pour the batter into the muffin cups, filling them ¾ full, and bake for 20-25 minutes or until a toothpick inserted into the center comes out clean.
Remove from the oven, let cool for 10 minutes, and turn out onto a wire rack to cool completely. Transfer to a baking sheet, carefully cover with plastic wrap and freeze overnight.
In a small saucepan, over medium-low heat, heat the apricot jam slightly. Add the lime juice and stir. (If the jam has small apricot pieces, you can use an immersion blender to undo them). With a kitchen brush, spread the jam around the sides and tops of the garibaldis and roll them in the sprinkles.