Make the cupcakes: Preheat the oven to 325 degrees F. Line a 12-cup muffin pan with white paper liners. Whisk the flour, baking powder, baking soda and salt in a medium bowl; set aside. Stir together the milk, lemon juice and vanilla in a small bowl; set aside (the milk will curdle). Beat the butter, 3/4 cup sugar and the lemon zest in a large bowl with a mixer on medium-high speed until light and fluffy, about 5 minutes. Reduce the mixer speed to low. Beat in the flour mixture in three batches, alternating with the milk mixture in two batches, until combined. Increase the mixer speed to medium and beat briefly until smooth.
Beat the egg whites in a large bowl with a mixer on high speed until foamy. Gradually beat in the remaining 1/4 cup sugar; continue beating until stiff glossy peaks form, about 3 minutes. Using a rubber spatula, fold the beaten egg whites into the batter in two batches. Divide the batter evenly among 6 small bowls; tint each with 1 to 3 drops gel food coloring, stirring to distribute the color.
Divide the purple batter evenly among the prepared muffin cups (about 1 1/2 teaspoons per cup) and gently spread to cover the bottom; layer the blue, green, yellow, orange and red batters on top, gently spreading each color but making sure not to swirl the colors together. (The liners will be full.) Bake until the tops of the cupcakes spring back when gently pressed, about 20 minutes. Let cool 5 minutes in the pan, then transfer to a rack to cool completely.
Meanwhile, make the frosting: Whisk the sugar, egg whites, water, vanilla, cream of tartar and salt in a large metal bowl set over a saucepan of simmering water until combined (do not let the bowl touch the water). With the bowl still over the simmering water, beat the mixture with a hand mixer on medium-high speed until stiff glossy peaks form, 5 to 7 minutes. Remove the bowl from the saucepan and let cool 5 minutes. Transfer the frosting to a piping bag fitted with a large round tip. Generously pipe onto the cupcakes.