In a large bowl, beat together the butter, salt, sugars, baking powder, and vanilla until well combined and smooth.
Add the eggs one at a time, beating well after each addition.
In a separate bowl, whisk together the sour cream or yogurt and milk till well combined.
Add the flour to the butter mixture alternately with the milk/sour cream mixture, beating gently to combine.
Pour/spread half the batter, roughly 1 1/3 cups, in each pan.
Sprinkle the filling evenly atop the batter.
Spread the remaining batter atop the filling. Use a table knife to gently swirl the filling into the batter, as though you were making a marble cake. Don't combine filling and batter thoroughly; just swirl the filling through the batter.
Bake the cake until it's a dark golden brown around the edges; medium-golden with no light patches showing on top, 50 to 55 minutes for the 8" round pans. When pressed gently in the middle, the cake should spring back. If the cake starts to get too brown, cover with aluminum foil for remaining cooking time.
Remove the cake from the oven and allow it to cool for 20 minutes before cutting and serving.