Melt 1/4 cup of butter in a saucepan, and stir in the confectioners' sugar and corn syrup until the sugar is dissolved. Bring to a boil over medium heat, stirring often, and stir in the pecans until well combined. Refrigerate the mixture for 30 minutes to chill.
Preheat oven to 350 degrees F (175 degrees C). Sift the flour and baking powder together in a bowl, and set aside.
Beat brown sugar, 3/4 cup butter, egg, and vanilla extract in a large bowl with an electric mixer on medium speed until the mixture is creamy, about 2 minutes. Gradually beat in the flour mixture until well mixed. Pinch off about 1 tablespoon of dough, and roll it into a ball. Press the dough into the bottom of an ungreased cupcake pan cup, and use your thumb to press the dough into a small piecrust shape, with 1/4-inch walls up the sides of the cupcake cup. Repeat with the rest of the dough. Fill each little crust with about 1 teaspoon of the prepared pecan filling.
Bake in the preheated oven until the cookie shells are lightly browned, 10 to 13 minutes. Watch closely after 10 minutes. Let the cookies cool in the cupcake pans for 5 minutes before removing to wire rack to finish cooling.