Beat softened cream cheese with a handheld mixer on medium-high speed for 30 seconds until smooth.
Reduce speed to medium-low and beat in powdered sugar and strawberry extract. Increase to medium-high speed and beat for another 30 seconds until smooth.
Scoop 24 dollops (about 1½ teaspoons each) of cream cheese mixture onto a parchment-lined baking sheet, spacing 2 inches apart. Gently press each dollop into flat disks. Freeze while preparing cookie dough.
In a medium bowl, whisk together flour, baking powder, and baking soda.
Beat softened butter on medium-high speed for 1 to 1½ minutes until smooth.
Add granulated and brown sugars to butter and mix for 1 minute until light and fluffy.
Mix in vanilla and strawberry extracts until just combined.
Add eggs one at a time, mixing well after each addition.
Lower mixer speed to low and gradually add the flour mixture, one cup at a time, mixing just until incorporated.
Toss chopped strawberries in 1 tablespoon of flour to coat, then gently fold into the cookie dough on lowest mixer speed.
Cover dough tightly and chill in the refrigerator for 1 hour.
Preheat oven to 375°F and line two baking sheets with parchment paper.
Scoop 24 large dollops of cookie dough (about 1½ teaspoons each) onto the prepared baking sheets, spacing 2 inches apart.
Remove cream cheese disks from freezer and place one disk in the center of each dough ball.
Press the cream cheese disks down gently, then use a small spatula or knife to lift and wrap cookie dough around the filling to encase it completely.
Bake cookies for 12 to 14 minutes until edges are golden brown.
Let cookies rest on the baking sheet for 10 to 12 minutes before transferring to a wire rack to cool completely.