Make the cupcakes: Preheat the oven to 325 degrees F. Line a 12-cup muffin pan with liners. Combine the butter, 2/3 cup condensed milk, the granulated sugar, egg, vanilla and salt in a food processor. Process until smooth and slightly thickened, about 2 minutes. Add the flour, baking powder and baking soda and pulse a few times until blended. Drizzle in 1/4 cup whole milk, pulsing a few times, until smooth.
Divide the batter among the prepared muffin cups, filling each about three-quarters of the way; sprinkle the mini chocolate chips on top. Bake until the tops of the cupcakes are golden and spring back when lightly pressed, 20 to 25 minutes. Let cool 5 minutes in the pan, then transfer to a rack. Poke the warm cupcakes all over with a toothpick. Mix the remaining 2 tablespoons each condensed milk and whole milk; brush evenly over the warm cupcakes. Let cool completely.
Make the frosting: Put the milk chocolate in a microwave-safe bowl and microwave in 30-second intervals, stirring, until smooth. Let cool slightly. Combine the butter, melted chocolate, 1/4 cup condensed milk and the salt in a food processor and process until smooth, scraping down the sides occasionally. Add the confectioners' sugar and process until smooth, about 1 more minute. Transfer to a bowl and refrigerate until firm but still spreadable, about 20 minutes.
Spread the frosting on the cupcakes, making a small well in the middle; fill each with 1/2 to 3/4 teaspoon condensed milk. Top with cookies.