Oil a grilling pan liberally with vegetable oil or nonstick cooking spray. Note: We used a grill pan so that if a ring broke we wouldn't lose it through the grates, but if you have thick slices, this should be less of an issue.
Preheat grill to medium high heat, about two minutes for a gas grill. Set the pan onto the grill so it preheats.
If not already pre-cut, slice pineapple into ¾” to 1" rings as evenly as possible. Set aside.
Melt butter in a microwave.
Combine butter, brown sugar, cinnamon, and lime in a small shallow bowl
Dredge both sides of the pineapple slices in the butter mixture or use a pastry brush to really drench each slice.
With grill tongs, add each slice to the hot grill pan. Set the timer for 7 minutes. Turn heat to high.
Leave the lid open while it grills. Turn the slices once and let grill for another 4 to 5 minutes. As it finishes cooking, dab any leftover butter mixture onto this side.
Remove pineapple to a clean plate with a spatula or grill tongs. Serve warm.