Line a small baking sheet or plate with parchment paper. In a small bowl, mix the cream cheese with powdered sugar until smooth and combined.
Place 10-12 heaping teaspoons of cream cheese mixture on the baking sheet or plate. Freeze for at least one hour, or until frozen almost completely solid.
When the cream cheese is frozen almost solid, line two large baking sheets with parchment paper and set them aside. Whisk together flour, salt, and baking soda then set this dry mixture aside.
In the bowl of a mixer, beat the butter and sugars together until light and fluffy, about 1 to 2 minutes. Mix in the egg, egg yolk, and vanilla extract.
Add the flour, salt, and baking soda mixture, and mix until just combined. Do not over-mix.
Dice the strawberries and gently fold them into the dough. Mix them in lightly to prevent the color from bleeding into the dough.
Scoop 12 heaping tablespoons of dough onto each baking sheet. Press the back of a teaspoon slightly into the top of half the balls of dough to create an indent.
Remove the frozen cream cheese filling from the freezer and place one into each of the indented cookies. Take a second ball of dough and place it over the cream cheese, pinching the sides to seal the cream cheese inside. Make sure not to squish the cream cheese into the cookie dough.
Wipe your hands clean before moving on to the next cookie. You should have tall balls of dough. Refrigerate the prepared cookie dough balls and repeat with the second cookie sheet and remaining dough.
Preheat the oven to 350°F and leave the cookie dough in the fridge while it preheats.
Bake the cookies in the preheated oven one sheet at a time for 13 to 17 minutes, or until the edges are slightly golden brown and the tops are set. Remove from the oven and leave them on the baking sheet for 10 to 15 minutes, then transfer to a cooling rack to cool completely.