Make brown butter: melt butter in saucepan over medium heat until frothy and brown specks appear (several minutes). Remove from heat and let cool.
Preheat oven to 350 degrees F. Line muffin pan with cupcake liners. Set aside.
In medium-sized bowl, mix together dry ingredients: flour, baking powder, and salt, stirring to combine. Set aside.
In a bowl attached to a stand mixer, beat cooled brown butter and sugar until combined (about 2 minutes), scrapping bowl as needed. Add eggs and vanilla and beat until light and fluffy, another 2-3 minutes.
With mixer on low (or by hand), gently mix in dry ingredients and milk alternately, starting and ending with dry ingredients until just combined.
Using large cookie scoop, scoop out dough into prepared pan and bake for 13-15 minutes or until toothpick inserted in middle comes out with only a few moist crumbs. Let cool in pan for 10 minutes then transfer onto wire rack to cool completely.
In bowl attached to a stand mixer, beat butter until smooth. Add molasses and beat until incorporated. Slowly add confectioners’ sugar and half and half and beat until light and fluffy, several minutes. Taste icing as needed to determine if more or less sugar is needed.