Place lemon cake mix into a large bowl and the strawberry cake mix into another large bowl. Mix 2 eggs, 1/3 cup vegetable oil, and 1 tsp lemon extract into each flavored cake mix until well blended.
Place dough in the refrigerator for 20 minutes to firm slightly.
Take 1 rounded teaspoon of lemon dough and roll into a ball. Do the same with the strawberry dough. Combine the lemon and strawberry dough to make one large ball.
Place the dough ball in a bowl with 1 cup of powdered sugar, and roll to lightly coat in sugar. Place on a parchment-covered baking sheet or Silpat mat at least 2 inches apart. Repeat with remaining dough.
Bake each batch 10 minutes until they turn slightly golden and are cooked through. Transfer to wire rack and let cool completely before serving.
To make the icing (optional), whisk together remaining cup powdered sugar with 1-2 tablespoons of milk until combined to desired consistency. Once the cookies reach room temperature, drizzle icing on each cookie with a fork or piping bag.
Make sure you place the dough in the refrigerator for at least 20 minutes. If the dough is too soft, it will stick to your hands and be hard to roll. Place back in the fridge between batches so dough stays firm.
If you'd like, you can whip up a batch of icing and drizzle over your cookies once they're cool, or just leave them plain, either way they're good!