Preheat the oven to 350°F.

Grease a 9 x 13-inch baking pan.

In a large bowl, whisk together the gluten-free flour, xanthan gum, salt, baking powder, and sugar.

In the bowl of a stand mixer, combine the eggs, egg yolks, oil, vanilla extract, and milk. Mix until well combined.

Gently sift the dry ingredients into the wet ingredients. Mix just until the batter is combined.

Pour the batter into the prepared 9 x 13-inch pan.

Bake for 40–45 minutes, or until a toothpick inserted into the center comes out clean.

Place the cake on a wire rack and allow it to cool completely.

While the cake cools, combine the halved strawberries and 3 tablespoons of sugar in a bowl. Use a potato masher to gently mash the strawberries, leaving some small pieces of strawberry throughout.

Spread the mashed strawberries evenly over the completely cooled cake.

Place the cake in the freezer for 20 minutes.

While the cake chills, make the cream cheese icing. Beat together the softened cream cheese and butter until smooth. Add the confectioner's sugar, vanilla extract, and milk. Beat until smooth and creamy.

Spread the cream cheese icing evenly over the strawberry layer.

Decorate the top of the cake with the halved strawberries.

Refrigerate until ready to serve.