Add the softened butter, granulated sugar, and brown sugar to a large mixing bowl. Using a hand or stand mixer, beat on medium-high speed for about 3 minutes, until the mixture becomes pale, light, and fluffy.
Add the egg and vanilla. Beat until completely incorporated, scraping the sides and bottom of the bowl as necessary.
In a separate bowl, whisk together the flour, cocoa powder, baking soda, and salt.
With the mixer running on low speed, gradually add the dry ingredients to the butter mixture. Mix only until a thick chocolate dough forms.
Add the milk and mix until incorporated. Fold or beat in the chocolate chips. At this stage, the dough will be thick and sticky.
Cover the bowl tightly and refrigerate the cookie dough for at least 3 hours. For easier shaping and deeper flavor, it can be refrigerated overnight or for up to 3 days.
Remove the chilled dough from the refrigerator and let it rest at room temperature for approximately 10 minutes.
Preheat the oven to 350°F (177°C). Line large baking sheets with parchment paper or silicone baking mats.
Measure about 2 tablespoons (40g) of cookie dough for each cookie and divide it into two pieces. Roll both pieces into balls.
Make an indentation in one piece and place a caramel in the center. Cover it with the second piece of dough and carefully seal the edges so the caramel is completely enclosed.
Repeat with the remaining dough. Arrange the stuffed cookie dough balls 2–3 inches apart on the prepared baking sheets.
Sprinkle each cookie lightly with flaky sea salt.
Bake the Salted Caramel Dark Chocolate Cookies for 12–13 minutes. The edges should look set while the centers remain visibly soft.
If the cookies have barely spread after about 9 minutes, carefully remove the baking sheet and tap it firmly against a heat-safe counter a few times. Return it to the oven to finish baking.
Let the cookies cool on the baking sheet for 5 minutes. For a bakery-style finish, gently press a few additional chocolate chips into the warm tops.
Transfer the cookies to a wire rack and allow them to finish cooling. The centers will settle slightly as they cool.