In a small mixing bowl, blend the softened cream cheese and powdered sugar until completely smooth with no lumps.
Scoop twelve heaping teaspoons of the cream cheese mixture onto a parchment-lined tray, spacing them out. Freeze for at least one hour until almost solid.
In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
In a large bowl or stand mixer, beat the softened butter, white sugar, and brown sugar on medium speed for about two minutes until pale, light, and fluffy.
Add the whole egg, egg yolk, and vanilla extract. Mix until smooth and fully combined.
Slowly add the dry flour mixture on low speed. Stop mixing the moment you see no more dry streaks to avoid tough cookies.
Using a rubber spatula, gently fold the diced, dried strawberries into the dough just until evenly spread. Do not use an electric mixer for this step.
Scoop twelve heaping tablespoons of dough onto two parchment-lined baking sheets. Use the back of a spoon to press a small indent into the center of each dough ball.
Place one frozen cream cheese dollop into each indent. Top with another heaping tablespoon of dough. Gently pinch the top and bottom pieces together to completely seal the filling inside.
Slide the baking sheets into the refrigerator to chill while you preheat the oven to 350 degrees F (175 degrees C).
Bake one sheet at a time on the middle rack for 13 to 17 minutes, until the edges turn a light golden brown. The tops should look set but still very soft.
Leave the cookies on the hot baking sheet for 10 to 15 minutes to finish baking, then transfer to a wire rack to cool completely.