Preheat the oven to 325 degrees F. Generously grease and lightly flour an 11x17-inch half-sheet pan or coat with baking spray (the floured kind, like this). Set aside.
In the bowl of a stand mixer fitted with the paddle attachment or a large bowl with a hand mixer or spoon, cream the butter for 1 minute on medium speed, until it is smooth and creamy. Add the sugar on medium speed and beat until fluffy and light in color. Beat in the honey, then the eggs and vanilla until smooth and nicely combined, stopping to scrape down the sides and bottom of the bowl a few times. The mixture may look curdled. Beat in the yogurt on medium-high speed until blended.
In a large bowl, whisk together the whole wheat flour, all-purpose flour, baking soda, and salt. Pour one-third of the flour mixture into the butter mixture. Beat on low speed for 5 seconds, then pour in half of the milk. Beat on low speed for 5 seconds. Repeat with the next third of the flour, then the remaining half of the milk. Add the last third of the flour and beat just until combined and the batter is smooth. Do not overmix.
Spread the cake batter into the prepared pan, smoothing it into a thin, even layer with a rubber spatula or offset spatula. Bake for 18 to 22 minutes or until the top is lightly browned and a toothpick inserted in the center comes out clean, rotating the pan 180 degrees once halfway through.
Place the baked cake on a wire rack and let cool in the pan completely. Meanwhile, make the frosting.
For the frosting: In the bowl of a stand mixer fitted with the paddle attachment or a large bowl with a hand mixer or spoon, beat the cream cheese, butter, powdered sugar, yogurt, and vanilla on medium speed until smooth and combined.
Spread the frosting over the cooled cake. Arrange blueberries in rows in top left corner to form the “star” portion of the flag. Line the strawberry slices in 5 rows to form the “stripes.” Refrigerate until ready to serve.
*You can substitute the same amount of all-purpose flour for the 1 1/4 cups white whole wheat flour. If you make this swap, reduce the amount of milk in the recipe to 3/4 cup plus 1 tablespoon.
Cake can be made 1 day in advance, covered tightly at room temperature, and then frosted before serving. Frosting can also be made 1 day ahead of time and stored in an airtight container in the refrigerator. Let come to room temperature for easier frosting of the cake.
Store leftovers in the refrigerator, tightly covered, for up to 3 days. You can also remove the fruit and store for up to 5 days in the refrigerator or freeze for up to 3 months.