Preheat oven to 350°F (175°C) and grease a 6-cavity mini bundt pan. Whisk flour, baking powder, and salt in a bowl.

Beat butter and sugar on medium speed until pale and fluffy, about 2 minutes.

Add eggs one at a time, beating 30 seconds each, then mix in the 2 teaspoons vanilla.

On low speed, add the flour mixture in three parts, alternating with buttermilk in two parts, mixing just until combined.

Spoon batter into prepared pan, filling each cavity two-thirds full. Bake for 13 to 15 minutes until a toothpick comes out with moist crumbs. Cool in pan 5 minutes, then turn out onto a rack to cool completely.

Whisk powdered sugar, 1 tablespoon milk, and ¼ teaspoon vanilla, adding more milk to reach a thick drizzling consistency. Drizzle over cooled cakes and let set 10 minutes.