In a large mixing bowl, combine the strawberry cake mix, 2 eggs, ? cup vegetable oil, and 1 teaspoon lemon juice. Mix until a soft dough forms.
In a separate mixing bowl, combine the lemon cake mix, remaining 2 eggs, remaining ? cup vegetable oil, and remaining 1 teaspoon lemon juice. Mix until a soft dough forms.
Cover both bowls and refrigerate the dough for 20–30 minutes.
Preheat the oven to 350°F. Line baking sheets with parchment paper or lightly grease them.
Scoop about 2 teaspoons of the lemon dough and roll it into a ball. Repeat with 2 teaspoons of the strawberry dough.
Gently press one lemon dough ball and one strawberry dough ball together, then roll them into one larger ball without completely blending the colors.
Place 1 cup powdered sugar in a shallow bowl. Roll each dough ball in the powdered sugar until well coated.
Arrange the cookies on the prepared baking sheets, spacing them about 2 inches apart.
Bake for 10–13 minutes, or until the edges are just beginning to turn lightly golden. Do not overbake.
Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely.
In a small bowl, whisk together 1 cup powdered sugar and the milk until smooth. Add a little more milk if needed to reach your desired drizzling consistency.
Drizzle the glaze over the cooled cookies using a fork or piping bag. Allow the glaze to set before serving.
Notes
Store the leftover cookies at room temperature in an airtight bag or container for up to 5 days. Keep a piece of bread in the container with the cookies to keep them soft.