Preheat the oven to 325°F with racks in top third and lower third positions. Line 2 large rimmed baking sheets with parchment paper; set aside. Gather all ingredients.
Beat brown sugar, butter, and granulated sugar with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Beat in egg and vanilla until well combined, about 30 seconds more.
Whisk together flour, instant espresso, baking powder, salt, and baking soda in a medium bowl until combined. Gradually add flour mixture to butter mixture, beating on low speed until just blended, 2 minutes.
Stir in semi-sweet chocolate chips, 1/2 cup of the bittersweet chocolate chips, and 1/2 cup of the milk chocolate.
Using a 1-ounce scoop or a 2-tablespoon measure, scoop dough about 2 inches apart onto prepared baking sheets (about 8 cookies per sheet).
Press remaining 1/4 cup bittersweet chocolate chips and 1/2 cup chopped milk chocolate onto each dough mound; reserve 1 cup remaining milk chocolate.
Bake in preheated oven until cookies start to spread, about 7 minutes. Dot cookies with remaining 1 cup milk chocolate. Rotate baking sheets from top to bottom and continue baking until lightly golden brown and centers are set, 8 more minutes.
Sprinkle with flaky salt.
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