Preheat the oven to 350F and line an 8x8 baking pan with parchment paper.
Place yogurt, eggs, oil, sugar and vanilla in a mixing bowl. Whisk together until evenly combined.
Pour all purpose flour, baking powder, baking soda and salt into the bowl and fold into wet ingredients with a flat spatula. Pour freeze dried strawberries into the bowl and mix in with the spatula.
Pour batter into prepared baking pan, smooth out and place in the oven. Bake at 350F for 40 minutes.
Remove from the oven and allow to cool completely before preparing the strawberry icing. Blend 1 cup of freeze dried strawberries into a powder.
Mix butter and cream cheese together with an electric mixer until evenly combined. Pour powdered sugar and strawberry powder into the bowl and mix again until evenly combine. Pour milk into the bowl and mix again until icing is soft and whipped. Spread icing over the cake, cut into 12 even pieces and serve.
Notes
Use freeze dried strawberries, not regular dried ones. Freeze dried strawberries blend into a fine powder and add intense flavor and a gorgeous pink color without adding moisture. Regular dried strawberries will not work the same way.
Make sure the butter and cream cheese are at room temperature. Cold butter and cream cheese will not mix together smoothly and you will end up with a lumpy frosting. Set them out at least 30 minutes before making the frosting.
Do not frost until the cake is completely cool. Even a slightly warm cake will cause the cream cheese frosting to melt. Wait at least one hour or chill the cake in the fridge to speed things up.
Blend the strawberries into a fine powder for the frosting. The finer the powder, the smoother the frosting will be. Use a blender, food processor, or spice grinder and blend until there are no visible pieces left.