Preheat oven to 350 degrees F. Line a 9-inch springform with parchment paper.
Add freeze dried strawberries to a food processor. Blend into a powder (this takes about one minute). There should be about 1 cup of strawberry powder.
In a large mixing bowl, combine flour, baking powder, kosher salt, and a ½ cup of the freeze dried strawberries. Whisk together and set aside.
In a separate bowl, melt butter. Then add sugar and vanilla extract. Whisk together for 1-2 minutes, then add egg and egg yolk. Continue to whisk for another minute, until eggs are fully combined.
Fold flour mixture to the wet ingredients until a thick batter forms. Add the batter to the springform pan and spread into an even layer.
Make the filling - add cream cheese and egg to a mixing bowl. Use hand mixer to mix together until smooth, about 1-2 minutes. Then add remaining strawberry powder, strawberry preserves, and 1 cup of powdered sugar at a time until smooth.
Pour filling on top of the cake batter. Bake for 45 minutes, until a brown skin forms on top of the cake. The center of the cake should be jiggly.
Allow the cake to cool at room temperature for 30 minutes, then place in the refrigerator to chill completely for about an hour.
Sprinkle with powdered sugar.