01 - Preheat oven to 350°F. Line two baking sheets with parchment paper and set aside.
02 - In a large mixing bowl, beat the softened butter and 1 cup of granulated sugar together using an electric mixer until pale, light, and fluffy, approximately 2-3 minutes. Add the egg and vanilla extract, continuing to mix until fully incorporated and smooth.
03 - In a separate medium bowl, whisk together the all-purpose flour, baking powder, and salt until evenly distributed. Gradually add the dry mixture to the wet ingredients in two additions, mixing on low speed just until the dough comes together. Avoid overmixing. Cover the bowl and refrigerate the dough for 20 minutes to firm up for easier handling.
04 - While the dough chills, combine the softened cream cheese, 1/3 cup granulated sugar, and 1 teaspoon vanilla extract in a small bowl. Beat with a mixer until silky smooth and free of lumps. Set aside.
05 - Roll tablespoon-sized portions of the chilled dough into balls and arrange them on the prepared baking sheets, spacing about 2 inches apart. Using your thumb or the back of a teaspoon, press a deep indentation into the center of each dough ball. Spoon approximately 1 teaspoon of the cheesecake filling into each well, smoothing the top gently.
06 - Bake at 350°F for 11 to 13 minutes, rotating the sheets halfway through, until the cookie edges are lightly golden and the centers appear set. Allow the cookies to rest on the baking sheets for 5 minutes before transferring to a wire cooling rack.
07 - Place the caramel candies and heavy cream in a microwave-safe bowl. Heat in 30-second intervals, stirring between each burst, until the caramel is completely melted and smooth. This typically takes 60-90 seconds total.
08 - Using a spoon or fork, drizzle the warm caramel sauce generously over each cooled cookie. For an elevated finish, sprinkle lightly with flaky sea salt. Allow the caramel to set at room temperature for 15-20 minutes before serving.