Beat the softened cream cheese until completely smooth, then mix in the powdered sugar, flour, and vanilla until thick and creamy. Scoop 16 rounded teaspoon-sized mounds onto a parchment-lined tray and freeze for at least 30 minutes, until firm.
Beat the softened butter and both sugars for 2 minutes until light and fluffy. Add the egg and vanilla, beat 1 minute, then mix in the whisked dry ingredients (flour, cornstarch, baking soda, salt) on low just until no dry streaks remain.
Cover the dough and refrigerate for 30 minutes while the filling finishes freezing. Line two rimmed baking sheets with parchment paper or silicone mats.
Scoop the chilled dough into 16 portions of about 2 tablespoons (35 g) each and roll into balls. Flatten each into a 3-inch disc, place a frozen filling mound in the center, fold the dough up and over, and pinch the seam completely closed. Roll smooth, place seam-side down on the baking sheet (8 per sheet, 2 inches apart), and chill 10 minutes if the dough gets sticky.
Preheat the oven to 350°F (175°C). Bake one sheet at a time on the center rack for 11 to 13 minutes, until the edges are golden and the centers look set but soft. Rest on the hot sheet for 5 minutes, then transfer to a wire rack to cool completely, about 30 minutes.
Microwave the unwrapped caramels with the heavy cream for 30 seconds, stir, then continue in 15-second bursts until smooth and pourable, about 1 1/2 to 2 minutes total. Reheat 10 seconds if it thickens while drizzling.
Drizzle the warm caramel over the cooled cookies in a zigzag pattern and immediately sprinkle each with a tiny pinch of flaky sea salt. Let sit 10 to 15 minutes so the caramel sets slightly before serving or stacking.