In a small bowl, mix the softened cream cheese, powdered sugar, and vanilla extract until smooth and creamy with no lumps.
Scoop into small teaspoon-sized portions, place on a parchment-lined plate, and freeze for 15 to 20 minutes until firm enough to handle. If they still feel soft, give them a few extra minutes so they’re easy to wrap later.
In a large bowl, cream the butter, brown sugar, and granulated sugar until light and fluffy, about 2 minutes with a mixer. Beat in the egg and vanilla extract until fully combined, scraping down the sides as needed.
Add the flour, baking soda, and salt, mixing just until a soft dough forms. Gently fold in half of the caramel sauce, leaving visible swirls. Chill the dough for 30 minutes. If it feels sticky after chilling, mix in 1 tablespoon of flour at a time until manageable.
Scoop the chilled dough into balls about 2 tablespoons each and flatten slightly in your palm.
Place a frozen cheesecake portion in the center and wrap the dough around it, sealing completely so no filling shows. Arrange on a parchment-lined baking sheet with space between each cookie, then drizzle or lightly swirl a little of the remaining caramel on top.
Preheat the oven to 350°F (175°C). Bake for 11 to 13 minutes, until the edges are set and lightly golden while the centers stay soft. Let the cookies cool on the baking sheet for 5 minutes before transferring to a cooling rack.
Once cooled, drizzle with extra caramel and finish with a light sprinkle of flaky sea salt if desired.