In a large bowl, cream together the softened butter, ¼ cup softened cream cheese, granulated sugar, and brown sugar until light and fluffy. Beat in the egg and 1 teaspoon vanilla extract until well combined.

In a separate medium bowl, whisk together the all-purpose flour, baking powder, and salt.

Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined, being careful not to overmix.

Cover and refrigerate the cookie dough for at least 30 minutes.

While the dough chills, beat 4 ounces softened cream cheese with powdered sugar and ½ teaspoon vanilla extract until smooth, then stir in the finely mashed or pureed strawberries. If needed, beat in the optional ½ teaspoon cornstarch if the filling seems too thin, and refrigerate until ready to use.

Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.

Drop rounded tablespoons of chilled cookie dough onto the prepared baking sheet, leaving about 2 inches between them. Place a small dollop (about ½ teaspoon) of the strawberry cheesecake filling on top of each cookie dough mound, and gently swirl the cheesecake filling into the cookie dough using a toothpick or small knife.

Bake for 10-12 minutes, or until the edges are lightly golden and the cheesecake filling appears mostly set.

Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely; the cheesecake filling will firm up further as it cools.