Heat a large griddle pan over medium-low heat. Spread the mayonnaise on one side of each of the bread. Place the bread, mayonnaise side down on the griddle and top with the slices of havarti.
Let melt slightly, about 2 minutes, then pile the lobster meat, cayenne, tarragon and sliced avocado on one of the slices and close the sandwich. Continue to cook until the cheese has melted and the bread is golden brown, about an additional 2 minutes on each side.
Remove and cut the sandwich in half then serve immediately.