Freeze the unwrapped caramel candies on a small plate for 10 minutes; this helps keep the centers gooey without leaking.
In a medium bowl, whisk together the flour, cocoa powder, baking soda, fine sea salt, and espresso powder until evenly combined.
In a large mixing bowl, beat the softened butter with the brown sugar and granulated sugar on medium speed until creamy and slightly lightened, about 2 minutes. Beat in the egg and vanilla just until smooth, 20–30 seconds.
Add the dry mixture to the wet ingredients and mix on low just until no dry streaks remain. Fold in the mini chocolate chips by hand. Cover and chill the dough for 15–20 minutes to make shaping easier.
While the dough chills, position a rack in the center of the oven and preheat to 355°F (180°C). Line 2 baking sheets with parchment paper.
Scoop rounded 1½-tablespoon portions of dough (about 25 g). Flatten each into a 2-inch disk in your palm, set a frozen caramel in the center, then wrap and pinch the dough to fully seal the candy. Place seam-side down on the prepared sheets, spacing about 2 inches apart. If dough softens, chill the tray for 5 minutes before baking.
Bake one sheet at a time for 9–11 minutes, until the edges are set and the tops lose their shine with fine cracks. For slightly crinklier tops, gently tap the baking sheet once on the oven rack at the 8-minute mark and finish baking.
Immediately after removing from the oven, sprinkle a pinch of flaky sea salt over each cookie. Cool on the sheet for 8 minutes (the caramel is very hot), then transfer to a rack to finish cooling. Enjoy warm or at room temperature.