Preheat oven to 350°F (175°C) and line baking sheets with parchment paper.
In a saucepan, cook strawberries and sugar over medium heat for 5 minutes until juicy. Stir in cornstarch and lemon juice, then cook until thickened. Let cool.
Beat cream cheese, sugar, and vanilla until smooth. Drop small portions onto parchment, freeze for 30 minutes.
Whisk flour, baking powder, baking soda, and salt. In another bowl, cream butter with both sugars until fluffy. Add egg, yolk, vanilla, and almond extract. Gradually stir in dry ingredients just until combined.
Scoop a tablespoon of dough, flatten, and place a frozen cheesecake ball in the center. Cover with more dough and seal edges. Top each cookie with strawberry sauce and swirl lightly.
Bake 11–13 minutes until edges are golden and centers soft. Cool 5 minutes before transferring to a rack.
Each cookie has chewy edges, a creamy cheesecake middle, and ribbons of fresh strawberry sweetness—perfect with milk or coffee.