Position your rack inside your oven to the lower third and preheat to 375 degrees. Grease and flour the wells of your mini bundt pan; tap out the excess flour.
Sift together the flour, baking powder and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium speed until creamy and smooth, about 30 seconds. Add the granulated sugar and continue beating until light and fluffy, about 4-5 minutes. Stop the mixer occasionally to scrape down the sides of the bowl. Add the eggs one at a time, beating well after each addition. Beat in the vanilla until well incorporated, about 1 minute.
Reduce the speed to low and add the flour mixture in three additions, alternating with the milk and beginning and ending with the flour. Beat each addition until just incorporated, again, stopping the mixer occasionally to scrape down the sides of the bowl.
Spoon about 2/3 cup of batter into each well and spread in the batter evenly. Bake until the cakes are lightly golden and a toothpick inserted in between the edge and the corner of the cakes comes out clean, for about 20-25 minutes. DO NOT OVERCOOK. When finished, transfer the pan to a wooden cutting board or other safe, cooling surface and let cool for about 10 to 15 minutes.
After this time, while the cakes are still a little warm, remove the cakes from the bundt pan and transfer to a wire rack on a cookie sheet. While the bundt cakes are still cooling, punch holes with a toothpick into the cake and drizzle very lightly or brush on the Kahlúa.