To make the cake:
Preheat oven to 350 degrees F. Have ready eight 8-ounce jars or ramekins.
To slice the pineapple: Using a very sharp knife, cut off the top and bottom of the fruit, so that it will sit flat on your cutting board. To remove the sides, start at the top and cut downward. Be sure you cut in deep enough to remove all of the hard spiny spots.
Once all of the exterior is removed, you will want to slice the pineapple in half, top to bottom. Slice the half into quarters and then remove the woody core of the fruit. Cut the cored piece in half again and then slice into 1/4-inch pieces.
Place the fruit into a pot and add the butter, sugar, rum and salt. Cook for 20 to 25 minutes to reduce the juices slightly. The timing will depend on the size of your pineapple. At the end of the cooking, add the thyme to the pineapples.
To mix the cake:
In a bowl whisk together the flour, baking powder and salt. Set aside. In a stand mixer, fitted with the paddle attachment, cream together the butter and brown sugar on medium speed, until light and fluffy. Add the eggs, one at a time, mixing until incorporated. Add half the flour mixture and mix just until blended. Mix in the yogurt, rum and vanilla. Add the remaining flour and blend well, be sure to scrape down the sides of the bowl.
Fill each of the jars or ramekins about a 1/4 full with pineapple and some of the juices. Divide the cake batter evenly over the pineapple. Add more pineapple and juice to the top, but just a thin layer. Bake for about 35 minutes or until cake is set. Serve warm or room temperature with a bit of lightly sweetened whipped cream or vanilla ice cream, and a sprig of thyme.
If you want to serve them as individual upside-down cakes, just run a knife around the outside edge of the jar and invert them onto a plate.