Preheat the grill to medium heat for direct grilling. I found that high heat created a little too much burn caramelization, so stick with medium...
Combine the butter, brown sugar and cinnamon in a small pot on the side burner of your grill (or stove inside).
Heat the butter, brown sugar and cinnamon over medium-low heat stirring constantly until all of the brown sugar has melted and then turn down to the lowest temperature. You can even turn the burner off at this point, we can warm it up when we are ready to brush onto the pineapple.
Place the pineapple slices or rings on the grill. Do not put the glaze on yet, we want to grill them just a bit first.
Grill the pineapple for about 2 minutes and then flip over to the other side.
At this point, warm up your glaze if it has solidified, stirring well to combine.
Now brush some of the glaze onto the pineapple slivers and close the grill's lid and grill for 2 minutes.
After 2 minutes, flip the pineapple over to the other side and again brush with glaze and close the lid for another 2 minutes.
Remove the pineapple from the grill and transfer to a bowl (a BIG bowl that will hold a lot of ice cream!).
Scope some of your favorite vanilla ice cream right on top of the warm pineapple and enjoy!