Preheat the oven to 350 F. Line a 12-cup muffin tin with cupcake liners.
Whisk together the flour, baking powder and salt in a medium bowl. Set aside.
In a large bowl, beat the eggs and granulated sugar with an electric mixer on medium speed until light and foamy, about 2 minutes. While beating, gradually pour in the butter and then the vanilla, mixing until combined.
Reduce the mixer speed to low and add half of the flour mixture, mixing until just combined. Add the milk, mixing until combined, and then the remaining flour mixture, mixing until just incorporated. Transfer one-third of the batter (about 1 cup) to a small bowl and stir in the cocoa powder. Alternate dropping spoonfuls of each batter into each cupcake liner, using a larger spoon for the white batter and a smaller spoon for the chocolate batter. Swirl the batters with an offset spatula or a skewer.
Bake until a toothpick inserted in the center of the cupcakes comes out clean, 18 to 20 minutes. Cool the cupcakes in the tin on a rack for 10 minutes, then remove from the tin and cool completely.
For the frosting:
Beat together the cream cheese and butter using an electric mixer on medium speed until fluffy, about 2 minutes. Beat in the vanilla and lemon juice. Reduce the mixer speed to low and gradually add the confectioners' sugar, beating until well combined.
Transfer the frosting to a piping bag or a resealable plastic bag and pipe the frosting onto the tops of the cupcakes.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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