Position a rack in the middle of the oven. Preheat the oven to 325° F. Line two baking sheets with parchment paper and butter the paper.

In a large bowl, using an electric mixer on low speed, mix the flour, brown sugar, and baking soda to blend them. Add the butter and continue mixing until the butter pieces are the size of peas, about 2 minutes; you will still see some loose flour. Stop the mixer and scrape the sides of the bowl as needed during mixing. Mix in the egg and vanilla. The batter will still look dry. Mix in the sour cream and milk until the batter looks evenly moistened. You may still see lumps of butter. Mix in the chocolate chips.

Drop heaping tablespoons (about 3 level tablespoons each) of dough at least 2½ inches apart onto the prepared baking sheets, Bake one sheet at a time until the tops feel soft but firm, the edges are lightly browned and crisp, and a toothpick inserted in the center comes out clean, about 17 minutes. (If the toothpick penetrates a chocolate chip, test another spot.)

Cool the cookies on the baking sheets for 10 minutes, then use a wide metal spatula to transfer them to a wire rack to cool completely.

THE FILLING:

In a medium saucepan, heat the cream and butter over low heat until the cream is hot and the butter melts. The hot cream mixture should form tiny bubbles and measure about 175° F on a thermometer; do not let the mixture boil. Remove the pan from the heat, add the chocolate chips, and let them sit in the hot cream mixture for about 30 seconds to soften. Whisk the filling until it is smooth and all of the chocolate is melted.

Pour the chocolate filling into a medium bowl and let it sit at room temperature until thickened enough to cling to the cookies, about 45 minutes. Or, refrigerate it until it thickens slightly, about 20 minutes.

Turn half of the cookies bottom side up. Leaving a 1/8-inch plain edge, Spread the filling over the bottoms of the cookies. If the filling is not thick and firm, let the cookies sit until the it firms up before adding the top cookie. Gently place the remaining cookies bottom side down on top of the chocolate filling.