First things first: I’ll leave it to @Hannibal to tell us how close this is to a real Kentucky Butter Cake (or, if there is even such a thing at all). Midwestern, mid-century recipes had a tendency to add all sorts of flourishes to their names in order to make them sound more interesting or exotic, when they were really nothing like what they claim to be. Case in point: any type of Mexican food recipe.
A caveat before we get started: I’ve never made a Bundt cake before. I’ve never had to. I did commercial and otherwise professional baking, and Bundt cakes were always viewed in my circle as something that was beneath us—something that home bakers did to try and be fancy. I will say this: it takes more practice and skill than my former colleagues gave it credit—mostly the actual turning over part. The making of the batter and baking of the cake is all pretty straightforward. But if the turning over part isn’t done just right it blows the whole thing. Fortunately for me I wasn’t making this for the local PTA meeting so it didn’t really matter.

Here’s what you’ll need:
Cake Batter
3 Cups Flour
2 Cups Sugar
1 tsp. Salt
1 tsp. Baking Powder
½ tsp. Baking Soda
1 Cup Buttermilk
1 Cup Butter, softened
4 Eggs
2 tsp. Vanilla or Rum Flavoring
Butter Sauce
¾ Cup Sugar
⅓ Cup Butter
3 Tbsp. Water
1-2 tsp. Vanilla or Rum Flavoring
To make the cake, in a large bowl, sift together flour, salt, baking powder and baking soda. In a separate bowl, cream together butter and sugar, adding sugar in a steady stream. Add eggs, one at a time, then buttermilk and flavoring, and beat until fluffy.

Add dry ingredients to wet ingredients. When combined, beat for an additional 3 minutes at medium speed.

To prepare your Bundt pan, use baker’s spray (not just any old cooking spray because baker’s spray has flour in it, which saves you a step) and coat the inside of the pan. Pour the batter into the Bundt pan and spread it evenly.

Bake in a 325-degree oven for 60-75 minutes.

To make the butter sauce, in a saucepan, heat sugar, water and butter until butter melts. Remove from heat. Add flavoring. I used the vanilla because that last thing the planet needs is me reacting to anything that tastes like booze.
When cake is fully baked, remove from oven. You’ll notice that it has developed a dome. Using a serrated knife, slice the dome off by laying the knife down on the edge of the Bundt pan to use as a leveling guide.

Pour ¾ cup of hot butter sauce over the cake. Cool in pan for five minutes.

Turn over onto a plate.

Spoon or carefully brush remaining sauce over cake.
Allow to cool completely, then serve at room temperature. Be sure to keep your ears open for any disparaging remarks.

That’s an interesting pattern on the bottom of the bundt pan and only adds to the appeal! Mine are smooth except for the scalloped one, which sometimes does weird things and is a pain to clean, so it is neglected but not unloved.
Maybe this is called a Kentucky Butter Cake because instead of the rum flavoring, as suggested here, you traditionally use bourbon, which they give the impression seems to run like tapwater in the Bluegrass State? I don’t know this first-hand, I’ve never been to Kentucky, for the record.
Adding liquid after baking is such a Bundt cake thing (yours looks quite good, btw.) I have recipes for lemon and cider cakes that also use the “ice pick” method.
Kentucky Butter Cake is indeed a thing. But I’ve never had it. I’ve seen it in stores and bakeries, and I love cake but don’t allow myself to eat it very often. I want to try this but considering the way I’ve been eating on vacation it’s going to have to wait until I get back into my healthy eating routine.