Whisk sugar, cornstarch, and kosher salt to combine in a small bowl.
Using an electric mixer on medium-high speed, beat cream cheese in a large bowl until smooth. Add sugar mixture and continue to beat until well combined. Fold in chocolate. Cover bowl with plastic wrap and freeze until very firm, at least 8 hours or up to overnight.
Place racks in top and bottom thirds of oven; preheat to 350°F. Line two rimmed baking sheets with parchment paper or silicone mats. Let dough sit at room temperature 5–10 minutes.
Using a spoon dipped in hot water, scoop 20 balls of dough (about 1 1/2 Tbsp. each) and roll into smooth balls with your hands. Using your thumb, make an imprint in center of ball, then fill each indent with about 1 tsp. cream cheese filling. Using your fingers, flatten edges of dough to about a 3" circle with the filling in the middle (like you’re making a dumpling). Fold dough over filling to cover completely; pinch to seal. Arrange on prepared sheets, spacing at least 3" apart. For best results, freeze dough 10 minutes.
Bake cookies, rotating pans halfway through, until edges are set and firm (the centers may be a little shiny and wet-looking), 12–14 minutes. Sprinkle cookies with sea salt while still warm, then let cool on baking sheets until firm. Transfer to a wire rack and let cool completely.