In a large mixing bowl, combine the strawberry cake mix, 2 eggs, 1/3 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 vegetable oil, and remaining 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 the 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 the glaze ingredients 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.