Preheat oven to 350F. Line a muffin tin for 12 cupcakes. In a bowl, whisk together flour, baking soda and salt. Set aside.
In a separate bowl, beat brown sugar and egg. Add vanilla and milk, mixing until combined. Gradually mix in melted butter, then add dry ingredients, stirring until smooth. Do not over-mix. Fold in chocolate chips.
Divide batter among cups and bake for 16-19 minutes, or until a toothpick inserted in the center comes out with only a few dry crumbs. Let cool in pan on a wire rack before removing.
For the frosting:
In a stand mixer fitted with paddle attachment, beat butter and sugars on medium speed until smooth and well combined. Add vanilla, salt and milk. Reduce speed to low and stir in flour. Whip until light and fluffy, adjusting liquid or flour as necessary to reach frosting consistency.
Fold in mini chocolate chips. [Note: skip this step if using regular sized chocolate chips and sprinkle them them on after frosting, unless tip of pastry bag is large enough for chips to pass through easily].
Pipe frosting onto cupcakes.
Notes
I recommend using mini chips if you want chips mixed into the frosting, otherwise use regular chips and sprinkle on top after frosting.