Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper.
Whisk flour, baking powder, baking soda, and salt in a medium bowl.
In a separate large bowl, cream butter, brown sugar, and granulated sugar until light and fluffy. Add egg and vanilla, mixing until smooth. Gradually stir in the dry ingredients until combined.
Beat cream cheese, powdered sugar, and vanilla extract in a small bowl until smooth and creamy.
Scoop about 1½ tablespoons of dough, flatten it slightly, and add 1 teaspoon of cheesecake filling in the center.
Seal the dough around the filling and place on the prepared baking sheet, spacing them 2 inches apart.
Bake for 10–12 minutes, or until edges are golden. The centers may be soft, but they'll firm up as they cool.
Let cookies cool for 5 minutes on the baking sheet, then transfer to a wire rack. Drizzle with caramel sauce and sprinkle with sea salt flakes. Allow the caramel to set before serving.