Prepare the Strawberry Sauce - In a small saucepan, combine strawberries and sugar over medium heat. Stir until the strawberries release their juices and soften, about 5 minutes. Add the cornstarch mixture and lemon juice, then cook for another 2-3 minutes until thickened. Remove from heat and cool completely. For a smoother texture, you can puree the sauce or leave it chunky for visible fruit pieces.
Make the Cheesecake Filling - Beat the cream cheese, sugar, and vanilla extract in a small bowl until smooth and creamy. Line a small tray with parchment paper and drop teaspoon-sized portions of the cheesecake mixture onto it. Freeze for at least 30 minutes to prevent the filling from leaking during baking.
Prepare the Cookie Dough - Whisk together flour, baking powder, baking soda, and salt in a medium bowl and set aside. In a large mixing bowl, beat the softened butter with granulated and brown sugars until light and fluffy, about 2-3 minutes. Add the egg, egg yolk, and extracts, mixing until well combined. Gradually incorporate the dry ingredients into the wet mixture, mixing until just combined to keep the dough soft and tender.
Assemble the Cookies - Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper. Take a tablespoon of cookie dough, flatten it slightly, then place a frozen cheesecake ball in the center. Cover with another tablespoon of cookie dough and seal the edges completely. Slightly flatten the top, then drizzle about 1/2 teaspoon of cooled strawberry sauce on top. Create a marble swirl pattern with a toothpick if desired.
Bake the Cookies - Bake for 11-13 minutes until the edges just start turning golden but centers still look slightly soft. Remove from the oven and let cool on the baking sheet for 5 minutes before transferring to a cooling rack to finish cooling completely.
Serve and Enjoy - Once cooled, enjoy these cookies with their chewy edges, soft centers, and creamy cheesecake filling bursting with strawberry flavor. They pair perfectly with cold milk or coffee.