Milk-and-Cookies Cupcakes

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For the cupcakes:

1 stick unsalted butter, at room temperature

2/3 cup plus 2 tablespoons sweetened condensed milk

1/4 cup granulated sugar

1 large egg

1 teaspoon pure vanilla extract

1/2 teaspoon salt

1 cup all-purpose flour

1 teaspoon baking powder

1/4 teaspoon baking soda

1/4 cup plus 2 tablespoons whole milk

2 tablespoons mini chocolate chips

For the frosting:

6 ounces milk chocolate, chopped

1 1/2 sticks (12 tablespoons) unsalted butter, at room temperature

1/4 cup sweetened condensed milk, plus more for topping

Pinch of salt

3/4 cup confectioners' sugar

Mini chocolate chip cookies, for topping